MAJHMATIKAIIteachers.cm.ihu.gr/anastasiou/wp-content/uploads/2012/02/notes... · xonec. Oi arijmoÐ (»;·;‡) apoteloÔn tic sunist¸sec tou monadiaÐou dianÔs-matoc ˆr. To eswterikì

ANWTATO TEQNOLOGIKO EKPAIDEUTIKO IDRUMA SERRWN

SQOLH TEQNOLOGIKWN EFARMOGWN

TMHMA PLHROFORIKHS & EPIKOINWNIWN

MAJHMATIKA IIApostìlhc KouiroukÐdhcEpisthmonikìc Sunergthc

SERRES 2005

2

AeÐ o Jeìc o Mègac GewmetreÐ(Pltwnac)

A.T.E.I Serr¸n c© Ap. KouiroukÐdhc Tm. Plhroforik c

Perieqìmena

1 Eisagwg 91.1 DianÔsmata, Sust mata Suntetagmènwn . . . . . . . . . . . . . 101.2 Analutik GewmetrÐa sto EpÐpedo kai ton Q¸ro . . . . . . . . 131.3 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Pragmatikèc Sunart seic 192.1 Sunart seic poll¸n metablht¸n . . . . . . . . . . . . . . . . . 202.2 Tìpoi, PedÐa OrismoÔ . . . . . . . . . . . . . . . . . . . . . . . 202.3 'Oria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4 Sunèqeia, Omoiìmorfh Sunèqeia . . . . . . . . . . . . . . . . . 252.5 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Parag¸gish 293.1 Merikèc Pargwgoi . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Armonikèc kai OmogeneÐc Sunart seic . . . . . . . . . . . . . . 323.3 SÔnjetec kai Peplegmènec Sunart seic . . . . . . . . . . . . . 333.4 Iakwbianèc, Diaforik . . . . . . . . . . . . . . . . . . . . . . 353.5 Polu¸numo Taylor . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Parag¸gish Oloklhr¸matoc . . . . . . . . . . . . . . . . . . . 373.7 Akrìtata, Akrìtata upì sunj kh . . . . . . . . . . . . . . . . 393.8 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Oloklhr¸mata 474.1 Dipl Oloklhr¸mata . . . . . . . . . . . . . . . . . . . . . . . 484.2 Tripl Oloklhr¸mata . . . . . . . . . . . . . . . . . . . . . . 504.3 Allag Metablht¸n se Pollapl Oloklhr¸mata . . . . . . . 514.4 Efarmogèc: Kèntra Mzac kai Ropèc Adrneiac . . . . . . . . 524.5 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Dianusmatik Anlush 575.1 Dianusmatikèc sunart seic mÐac metablht c . . . . . . . . . . 585.2 Dianusmatikèc sunart seic poll¸n metablht¸n . . . . . . . . . 595.3 KlÐsh, Apìklish, Strof . . . . . . . . . . . . . . . . . . . . . 605.4 EpikampÔlia kai Epiepifneia Oloklhr¸mata . . . . . . . . . . 63

3

4 PERIEQOMENA

5.5 Ta Jewr mata Green kai Gauss . . . . . . . . . . . . . . . . . 655.6 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Diaforikèc Exis¸seic 716.1 Eisagwg stic Diaforikèc exis¸seic . . . . . . . . . . . . . . 726.2 Diaforikèc Exis¸seic Qwrizomènwn Metablht¸n . . . . . . . . 756.3 OmogeneÐc D.E. pr¸thc txhc . . . . . . . . . . . . . . . . . . 756.4 Grammikèc D.E. pr¸thc txhc . . . . . . . . . . . . . . . . . . 766.5 Grammikèc D.E. deÔterhc txhc . . . . . . . . . . . . . . . . . . 776.6 Arijmhtik lÔsh D.E. pr¸thc txhc . . . . . . . . . . . . . . . 786.7 Diaforikèc exis¸seic me merikèc parag¸gouc . . . . . . . . . . 796.8 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7 MigadikoÐ ArijmoÐ 857.1 Eisagwg stouc MigadikoÔc ArijmoÔc . . . . . . . . . . . . . . 867.2 Polik kai Trigwnometrik Morf . . . . . . . . . . . . . . . . 887.3 RÐzec Migadik¸n Arijm¸n . . . . . . . . . . . . . . . . . . . . 897.4 Migadik strefìmena dianÔsmata . . . . . . . . . . . . . . . . 907.5 Ask seic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Aþ Kampulìgramma Sust mata Suntetagmènwn 95

Bþ To trÐakmo tou Frenet 99


Katlogoc Sqhmtwn

1.1 H sunrthsh antÐstrofh efaptomènh. . . . . . . . . . . . . . . 121.2 H sunrthsh antÐstrofo sunhmÐtono. . . . . . . . . . . . . . . 13

3.1 Sqèsh sunèqeiac kai paragwgisimìthtac . . . . . . . . . . . . 31

Aþ.1 SÔsthma Kulindrik¸n Suntetagmènwn . . . . . . . . . . . . . . 96Aþ.2 SÔsthma Sfairik¸n Suntetagmènwn . . . . . . . . . . . . . . . 96

5

6 KATALOGOS SQHMATWN


Katlogoc Pinkwn

1.1 Oi kwnikèc tomèc. . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 PeriorismoÐ sto pedÐo orismoÔ. . . . . . . . . . . . . . . . . . . 22

3.1 Qarakthrismìc akrottwn. . . . . . . . . . . . . . . . . . . . . 40

6.1 Yeudok¸dikac gia thn mèjodo Euler . . . . . . . . . . . . . . . 79

7

PROLOGOSOi shmei¸seic1 autèc kalÔptoun thn Ôlh tou maj matoc Majhmatik II

pou didsketai sto B' exmhno tou tm matoc Plhroforik c & Epikoinwni¸n,tou T.E.I. Serr¸n. Sto pr¸to keflaio (Eisagwg ) gÐnetai mÐa sÔntomhanaskìphsh basik¸n ennoi¸n apì thn GewmetrÐa. To keflaio autì èqeiskopì na apotelèsei anafor gia kpoiec basikèc ènnoiec pou qreizontaiparaktw, par na knei exantlhtik parousÐash tou jèmatoc thc Analu-tik c GewmetrÐac. Sta epìmena keflaia parousizontai oi sunart seic, dÔo perissotèrwn metablht¸n, h diaforisimìtht touc, kai h oloklhrwsimìthttouc (dipl & tripl oloklhr¸mata). EpÐshc meletme thn basik dianus-matik anlush pou eÐnai aparaÐthth kai se jèmata Fusik c.

Paraktw gÐnetai mÐa eisagwg stic sun jeic diaforikèc exis¸seic pr¸thckai deÔterhc txhc, mazÐ me kpoiec stoiqei¸deic mejìdouc arijmhtik c epÐlus ctouc. Tèloc eisgontai oi migadikoÐ arijmoÐ, mazÐ me tic basikèc efarmogèctouc se probl mata Hlektrik¸n kulwmtwn.

Sunisttai anepifÔlakta stouc spoudastèc na knoun eureÐa qr sh an¸ter-wn glwss¸n programmatismoÔ (ìpwc C ++) /kaÐ sqediastik¸n pakètwn giathn grafik parstash twn sunart sewn poll¸n metablht¸n, kaj¸c kai twnapotelesmtwn apì thn arijmhtik epÐlush twn diaforik¸n exis¸sewn. Sekje keflaio h kaluptìmenh Ôlh epibllei grafikèc parastseic pou oxÔ-noun thn skèyh kai apaitoÔntai gia thn kal katanìhs thc. H sundiasmènhqr sh anwtèrwn glwss¸n programmatismoÔ kai programmtwn grafik¸n stajèmata pou pragmateuìmaste ed¸, apoteleÐ exairetik qr simh praktik .

Apìstoloc KouðroukÐdhc

1 c© Dr. Apìstoloc KouiroukÐdhc

Keflaio 1

Eisagwg

9

10 KEFALAIO 1. EISAGWGH

1.1 DianÔsmata, Sust mata SuntetagmènwnSton trisdistato EukleÐdio q¸ro R3 èna shmeÐo P mporeÐ na oristeÐ monos -manta apì tic kartesianèc suntetagmènec tou P (x, y, z), wc prìc èna trisor-jog¸nio sÔsthma axìnwn Oxyz. To dinusma jèshc tou shmeÐou eÐnai

~r = xx̂ + yŷ + zẑ (1.1.1)

ìpou ta {x̂, ŷ, ẑ} eÐnai ta monadiaÐa dianÔsmata (me mètro monda) kat m koctwn axìnwn. UpenjumÐzoume ìti gia èna dinusma ~A = (A1, A2, A3) to mètrotou dÐnetai apì thn

| ~A| =√

A21 + A22 + A

23 (1.1.2)

OrÐzontai epÐshc kai ta sunhmÐtona kateÔjunshc, wc

ξ := cosφ1 =x√

x2 + y2 + z2

η := cosφ2 =y√

x2 + y2 + z2

ζ := cosφ3 =z√

x2 + y2 + z2(1.1.3)

ìpou (φ1, φ2, φ3) eÐnai oi gwnÐec pou sqhmatÐzei to dinusma ~r me touc treÐcxonec. Oi arijmoÐ (ξ, η, ζ) apoteloÔn tic sunist¸sec tou monadiaÐou dianÔs-matoc r̂.

To eswterikì ginìmeno dÔo dianusmtwn orÐzetai wc

~A · ~B = | ~A|| ~B|cosφ (1.1.4)

ìpou φ eÐnai h metaxÔ touc gwnÐa. En dÔo dianÔsmata eÐnai kjeta, toeswterikì touc ginìmeno eÐnai mhdèn. IsqÔei epÐshc ~a · ~a = |~a|2 en¸ se sqèshme tic sunist¸sec touc, to eswterikì ginìmeno dÔo dianusmtwn dÐnetai apì

~A · ~B = A1B1 + A2B2 + A3B3 (1.1.5)

Tèloc h probol tou ~A pnw sto ~B dÐnetai apì thn

Π ~B(~A) = ~A · ~B

~B

| ~B|2(1.1.6)

To exwterikì ginìmeno dÔo dianusmtwn orÐzetai wc

~A× ~B = | ~A|| ~B|sinφ n̂ (1.1.7)


1.1. DIANUSMATA, SUSTHMATA SUNTETAGMENWN 11

ìpou to monadiaÐo n̂ eÐnai kjeto sto epÐpedo twn ~A, ~B kai h for tou ka-jorÐzetai apì ton kanìna tou dexioÔ qerioÔ. Se sunrthsh me tic sunist¸sectouc, to exwterikì ginìmeno dÔo dianusmtwn dÐnetai apì thn

~A× ~B =∣∣∣∣∣∣

x̂, ŷ, ẑA1, A2, A3B1, B2, B3

∣∣∣∣∣∣= x̂(A2B3 − A3B2) +

+ ŷ(A3B1 − A1B3) + ẑ(AxBy − AyBx) (1.1.8)

EpÐshc to mètro tou exwterikoÔ ginomènou | ~A× ~B| = EAB, dhl. mac dÐnei toembadìn tou parallhlogrmmou pou sqhmatÐzoun ta dÔo dianÔsmata. Tèlocan ta dianÔsmata eÐnai suggrammik tìte ~A× ~B = 0.

To meiktì ginìmeno tri¸n dianusmtwn orÐzetai wc

( ~A, ~B, ~C) := ~A · ( ~B × ~C) =∣∣∣∣∣∣

A1, A2, A3B1, B2, B3C1, C2, C3

∣∣∣∣∣∣(1.1.9)

H apìluth tim tou meiktoÔ ginomènou mac dÐnei ton ìgko tou parallhlepipè-dou pou orÐzetai apì ta trÐa dianÔsmata. Etsi an trÐa dianÔsmata eÐnaisunepÐpeda to meiktì touc ginìmeno mhdenÐzetai, en¸ en eÐnai diforo toumhdenìc shmaÐnei ìti ta trÐa dianÔsmata mporoÔn na apotelèsoun bsh, dhl.kje llo dinusma na analujeÐ wc proc aut (na grafeÐ wc grammikìc sun-diasmìc touc).

Ektìc apì to trisorjog¸nio kartesianì sÔsthma suntetagmènwn, upr-qoun kai lla sust mata suntetagmènwn me eidik summetrÐa pou pollècforèc eÐnai qr sima se sugkekrimèna probl mata. Oi kulindrikèc suntetag-mènec (r, θ, z) orÐzontai gia kje shmeÐo ektìc thc arq c O(0, 0, 0) kai

x = rcosθ

y = rsinθ

z = z, r > 0, 0 ≤ θ < 2π (1.1.10)Gia thn arq èqoume r = z = 0 all to θ eÐnai aprosdiìristo. EpÐshc oiantÐstrofoi metasqhmatismoÐ dÐnontai apì thn

r =√

x2 + y2

θ =

Arctan(y/x), x > 0, y > 0π + Arctan(y/x) x < 02π + Arctan(y/x) x > 0, y < 0

θ = (π/2) x = 0, y > 0θ = (3π/2) x = 0, y < 0

(1.1.11)

ìpou me V (u) = Arctan(u) sumbolÐzoume ton prwteÔonta kldo thc an-tÐstrofhc efaptomènhc pou faÐnetai sto Sq ma 1.1.



Sq ma 1.1: H sunrthsh V (u) = Arctan(u).

Tèloc ta monadiaÐa dianÔsmata pou deÐqnoun kat thn dieÔjunsh pouauxnontai oi kulindrikèc suntetagmènec dÐnontai apo tic

r̂ = cosθ x̂ + sinθ ŷ

θ̂ = −sinθ x̂ + cosθ ŷẑ = ẑ (1.1.12)

O stoiqei¸dhc ìgkoc stic kulindrikèc suntetagmènec dÐnetai apì thn dV =rdrdθdz en¸ h stoiqei¸dhc epifneia pnw se kÔlindro aktÐnac R apì thndS = Rdθdz.

Oi sfairikèc suntetagmènec (r, θ, φ) orÐzontai gia kje shmeÐo ektìc thcarq c O(0, 0, 0) apì tic

x = rsinθcosφ

y = rsinθsinφ

z = rcosθ (1.1.13)

ìpou r > 0, 0 ≤ θ ≤ π kai 0 ≤ φ < 2π. H arq èqei r = 0 all ta θ, φ eÐnaiaprosdiìrista. Oi antÐstrofoi metasqhmatismoÐ dÐnontai apì

r =√

x2 + y2 + z2

θ = Arccos(z/r)

φ =

Arctan(y/x), x > 0, y > 0π + Arctan(y/x) x < 02π + Arctan(y/x) x > 0, y < 0

θ = (π/2) x = 0, y > 0θ = (3π/2) x = 0, y < 0

(1.1.14)


1.2. ANALUTIKH GEWMETRIA STO EPIPEDO KAI TON QWRO 13

ìpou me V (u) = Arccos(u) : [−1, 1] → [0, π] sumbolÐzoume ton prwteÔontakldo thc sunrthshc tou antÐstrofou sunhmitìnou pou faÐnetai sto Sq ma1.2.

Sq ma 1.2: H sunrthsh V (u) = Arccos(u).

Ta monadiaÐa dianÔsmata sundèontai me tic sqèseic

r̂ = sinθcosφ x̂ + sinθsinφ ŷ + cosθ ẑ

θ̂ = cosθcosφ x̂ + cosθsinφ ŷ − sinθ ẑφ̂ = −sinφ x̂ + cosφ ŷ (1.1.15)

O stoiqei¸dhc ìgkoc stic sfairikèc suntetagmènec dÐnetai apì thn dV =r2sinθdrdθdφ en¸ h stoiqei¸dhc epifneia pnw se sfaÐra aktÐnac R apì thndS = R2sinθdθdφ.Ta parapnw sust mata suntetagmènwn kai ta monadiaÐa touc dianÔsmatafaÐnontai sto Parrthma A'.

1.2 Analutik GewmetrÐa sto EpÐpedo kai tonQ¸ro

MÐa kampÔlh sto epÐpedo perigrfetai genik apì mÐa sqèsh thc morf cF (x, y) = 0. En mporèsoume na epilÔsoume thn sqèsh aut wc prìc y =f(x) tìte èqoume kampÔlh sto epÐpedo pou eÐnai kai h grafik parstashthc sunrthshc f . MÐa isodÔnamh èkfrash gia mÐa kampÔlh, dosmènh separametrik morf , eÐnai h

x = x(t),

y = y(t), t ∈ [a, b]. (1.2.1)



Gnwstèc kampÔlec sto epÐpedo eÐnai oi legìmenec kwnikèc tomèc, pou eÐnaideutèrou bajmoÔ wc prìc tic suntetagmènec (x, y). Otan grafoÔn se katllh-lo sÔsthma suntetagmènwn lambnoun mÐa apì tic morfèc pou faÐnontai stonPÐnaka 1.1. Ed¸ R > 0 eÐnai h aktÐna tou kÔklou kai a, b > 0 oi hmixonecthc èlleiyhc kai thc uperbol c antÐstoiqa.

KÔkloc x2 + y2 = R2

'Elleiyh x2a2

+ y2

b2= 1

Parabol y2 = 2px, x2 = 2py

Uperbol x2a2− y2

b2= ±1

PÐnakac 1.1: Oi kwnikèc tomèc.

Ston q¸ro mÐa epifneia perigrfetai genik apì mÐa sqèsh thc morf cG(x, y, z) = 0. En mporèsoume na epilÔsoume thn sqèsh aut wc prìc z =g(x, y) tìte èqoume ston q¸ro mÐa epifneia pou eÐnai kai grafik parstashthc en-lìgw sunrthshc dÔo anexrthtwn metablht¸n. IsodÔnama mporoÔmena gryoume mÐa epifneia sto q¸ro se parametrik morf wc

x = x(t, s)

y = y(t, s)

z = z(t, s), (t, s) ∈ R2 (1.2.2)MporoÔme sta parapnw na antiparablloume to gegonìc ìti mÐa kampÔlh s-ton q¸ro dÐnetai genik se parametrik morf wc ~r = ~r(t) = (x(t), y(t), z(t)), t ∈R.

MÐa shmantik kathgorÐa epifanei¸n ston q¸ro eÐnai oi epifneiec deutèr-ou bajmoÔ oi opoÐec se katllhlo sÔsthma suntetagmènwn lambnoun mÐa apìtic paraktw morfèc:

• Elleiyoeidècx2

a2+

y2

b2+

z2

c2= 1 (1.2.3)

• Monìqwno Uperboloeidècx2

a2+

y2

b2− z

2

c2= 1 (1.2.4)


1.2. ANALUTIKH GEWMETRIA STO EPIPEDO KAI TON QWRO 15

• DÐqwno Uperboloeidèc

x2

a2+

y2

b2− z

2

c2= −1 (1.2.5)

• K¸nocx2

a2+

y2

b2= z2 (1.2.6)

• Elleiptikì Paraboloeidèc

x2

a2+

y2

b2= 2γz (1.2.7)

• Uperbolikì Paraboloeidèc

x2

a2− y

2

b2= 2γz (1.2.8)

• Parabolikìc KÔlindroc

y2 = 2px (1.2.9)

• Elleiptikìc KÔlindroc

x2

a2+

y2

b2= 1 (1.2.10)

• Uperbolikìc KÔlindroc

x2

a2− y

2

b2= 1 (1.2.11)

• EpÐpedoAx + By + Cz + D = 0 (1.2.12)

H apìstash dÔo shmeÐwn A(A1, A2, A3) kai B(B1, B2, B3) dÐnetai apì thnsqèsh

d(A,B) =

[3∑

i=1

(Ai −Bi)2]1/2

(1.2.13)

en¸ h apìstash tou shmeÐou P1(x1, y1, z1) apì to epÐpedo thc Ex. (1.2.12)apì thn

d =|Ax1 + By1 + Cz1 + D|√

A2 + B2 + C2(1.2.14)



MÐa kampÔlh ston q¸ro mporeÐ epÐshc na ekfrasteÐ wc h tom dÔo epi-fanei¸n, G1(x, y, z) = 0 kai G2(x, y, z) = 0. H exÐswsh t¸ra thc eujeÐacpou dièrqetai apì to shmeÐo P1(x1, y1, z1) kai eÐnai parllhlh prìc to dinus-ma ~a = (a1, a2, a3) dÐnetai apì thn

x− x1a1

=y − y1

a2=

z − z1a3

, |~a| 6= 0 (1.2.15)

en¸ h eujeÐa pou dièrqetai apì dÔo shmeÐa (P1 6= P2) dÐnetai apì thn sqèshx− x1x2 − x1 =

y − y1x2 − y1 =

z − z1z2 − z1 (1.2.16)

Tèloc mÐa eujeÐa mporeÐ na oristeÐ wc h tom twn dÔo epipèdwn Aix + Biy +Ciz+Di = 0, (i = 1, 2). Ac orÐsoume a = (B1C2−C1B2), b = (C1A2−A1C2)kai c = (A1B2 − A2B1) kai èstw ìti (qwrÐc blbh thc genikìthtac) c 6= 0.Tìte h parametrik morf thc eujeÐac mporeÐ na grafeÐ wc

x = x1 + ta

y = y1 + tb

z = tc, t ∈ R (1.2.17)ìpou ta (x1, y1) eÐnai lÔsh tou sust matoc

A1x1 + B1y1 + D1 = 0

A2x1 + B2y1 + D2 = 0 (1.2.18)

1.3 Ask seic

1. DÐnontai ta dianÔsmata ~v1 = 1√33(2x̂ + 2ŷ + 5ẑ), ~v2 =1√3(x̂ + ŷ + ẑ),

~v3 =1√11

(3x̂ + ŷ + ẑ). Na apodeiqjeÐ ìti apoteloÔn monadiaÐa bsh kaina analujeÐ wc prìc aut th bsh to ~a = 3x̂ + 5ŷ + 4ẑ.

2. DÐnetai to shmeÐo P (−1,−4,−6). Na brejoÔn ta shmeÐa P1, P2, P3 kaiP4 ta opoÐa eÐnai summetrik tou P wc prìc ta epÐpeda Oxy, Oyz, Ozxkai thn arq O tou Oxyz antÐstoiqa.

3. DÐnontai ta shmeÐa B, G pnw ston xona Ox me tetmhmènh 2 kai λ2 −10λ + 23 ìpou λ ∈ R. Na brejeÐ to l ¸ste to dinusma BG na èqeimètro monda kai h tetmhmènh tou shmeÐou G.

4. Na brejoÔn ta sunhmÐtona kateÔjunshc, ta mètra kai oi gwnÐec me toucxonec tou sust matoc Oxyz twn dianusmtwn 1√

3(1, 1, 1) kai (1, 0,−1).

EpÐshc na brejeÐ h metaxÔ touc gwnÐa.


1.3. ASKHSEIS 17

5. Na deÐxete ìti ta dianÔsmata ~a(2,−3,−1) kai ~b(−6, 9, 3) eÐnai parllh-la. Na brejeÐ to embadìn Eab tou parallhlogrmmou pou sqhmatÐzounta ~a(1, 0, 0) kai ~b(2, 2, 0).

6. Gia dÔo sunepÐpeda dianÔsmata ~a, ~b deÐxte ìti h meglh δ1 kai h mikr δ2 diag¸nioc tou parallhlogrmmou pou sqhmatÐzoun dÐnontai apì tÐc

δ21 = |~a|2 + |~b|2 + 2|~a||~b|cosφδ22 = |~a|2 + |~b|2 − 2|~a||~b|cosφ

ìpou φ h metaxÔ touc gwnÐa.

7. DeÐxte ìti ta ~a(1, 1, 1), ~b(2, 2, 2) kai ~c(3, 4, 5) eÐnai sunepÐpeda. NaanalujeÐ to èna apì aut se sqèsh me ta lla dÔo.

8. Na brejeÐ mÐa trida apì ta ~a(5, 6, 1),~b(3, 1, 2), ~c(1,−1, 1) kai ~d(4, 5,−1)pou sqhmatÐzei bsh. Na analujeÐ to llo dinusma wc prìc aut n.

9. Na brejeÐ dinusma monadiaÐo kai kjeto sta v1(1,−1, 1) kai ~v2(1, 1,−1).

10. DeÐxte tic paraktw tautìthtec

~a×~b = −~b× ~a~a× ~a = 0

~a · (~b× ~c) = ~c · (~a×~b) = ~b · (~c× ~a)~a× (~b× ~c) = (~a · ~c)~b− (~a ·~b)~c

(~a×~b)× (~c× ~d) = (~a,~b, ~d)~c− (~a,~b,~c)~d

11. Na breÐte tic suntetagmènec tou P (1, 1, 1) stic kulindrikèc kai sfairikècsuntetagmènec.

12. Na epilujoÔn oi sqèseic (1.1.12) kai (1.1.15) wc proc ta monadiaÐa di-anÔsmata twn kartesian¸n suntetagmènwn kai na grafeÐ to dinusmajèshc thc Ex. (1.1.1) stic antÐstoiqec suntetagmènec. Efarmog : ToshmeÐo P (1, 1, 0).

13. Na grafoÔn parametrikèc exis¸seic gia ton kÔklo, thn èlleiyh, thnparabol y2 = 2px kai thn uperbol , wc epÐpedec kampÔlec.

14. Estw h èlleiyh kai E ′(−c, 0) (c2 = a2 − b2) h estÐa thc pou brÐsketaisto arnhtikì tm ma tou xona Ox. Jewr¸ntac to shmeÐo autì arq sust matoc polik¸n suntetagmènwn, jèste x + c = rcosθ, y = rsinθ



sthn kartesian exÐswsh thc èlleiyhc kai breÐte thn polik thc exÐswshwc

r =(b2/a)

1− ecosθìpou e = (c/a) h ekkentrìtht thc. An o Hlioc eÐnai sthn estÐa thcèlleiyhc tìte h troqi thc Ghc perigrfetai se polikèc suntetagmènecapì aut thn sqèsh.

15. Na brejeÐ h apìstash metaxÔ twn shmeÐwn O(0, 0, 0) kai (acost, bsint, 0),t ∈ [0, 2π). Gia poi shmeÐa gÐnetai mègisth; Elqisth;

16. DeÐxte ìti to dinusma ~V (A,B,C) eÐnai kjeto sto epÐpedo Ax+By +Cz+D = 0. BreÐte thn exÐswsh tou epipèdou pou pernei apì ta shmeÐaP (0,−2, 0), Q(−1,−2,−3) kai eÐnai kjeto sto epÐpedo 3x + 2y + 6z−6 = 0.

17. BreÐte to epÐpedo pou dièrqetai apì ta shmeÐa A(1, 0, 0), B(0, 2, 0) kaiC(0, 0, 3). Poi eÐnai h dianusmatik monda pou eÐnai kjeth s' autì;Na brejeÐ h apìstash thc arq c twn axìnwn apì to epÐpedo.

18. DÐnontai ta dianÔsmata ~a(3, 4, 5) kai ~b(0, 6, 1). Na grafeÐ h exÐswsh thceujeÐac pou pernei apì thn arq twn axìnwn kai eÐnai parllhlh prìcto ~a ×~b. Na grafeÐ epÐshc h exÐswsh tou epipèdou pou dièrqetai apìthn parapnw eujeÐa kai eÐnai parllhlo prìc to dinusma ~c(1, 1, 1).

19. Estw h eujeÐa pou orÐzetai wc tom twn epipèdwn 2x + 3y + z − 5 = 0kai 6x + 7y + 8z + 6 = 0. Na grafoÔn oi parametrikèc thc exis¸seic.EpÐshc na brejeÐ h exÐswsh thc eujeÐac pou eÐnai parllhlh proc thnparapnw kai dièrqetai apì to shmeÐo tom c twn epipèdwn x + y = 4,y + z = 6 kai z + x = 12.

20. Na upologisteÐ me olokl rwsh o ìgkoc kai h epifneia miac sfaÐracaktÐnac R.


Keflaio 2

Pragmatikèc Sunart seic

19

20 KEFALAIO 2. PRAGMATIKES SUNARTHSEIS

2.1 Sunart seic poll¸n metablht¸n

'Otan èna metablhtì mègejoc exarttai apì lla metablht megèjh (dÔo perissìtera) tìte orÐzetai mÐa sunrthsh poll¸n metablht¸n. San pardeigmamporoÔme na anafèroume ton ìgko enìc idanikoÔ aerÐou, pou apoteleÐ sunrthshdÔo metablht¸n V = V (p, T ) = nRT/p, dhl. thc pÐeshc tou kai thc jermokrasÐ-ac. Ed¸ to n =arijmìc moles kai to R = 0.082 litr·atm/grad·moles nooÔntaiwc stajerèc.

H melèth thc fusiologÐac enìc anjr¸pou mac deÐqnei ìti h epifneia tous¸matìc tou S(m2) se t. mètra exarttai apì to broc tou W (Kg) sekil kai to Ôyoc tou H(cm) se ekatost mèsw thc sqèshc S = S(W,H) =0.007184W 0.425H0.725.

Tèloc o nìmoc thc pagkìsmiac èlxhc tou NeÔtwna anmesa se dÔo mzecM1,M2 dÐnetai apì thn sqèsh F (x, y, z) = −GM1M2/(x2 + y2 + z2). Taparapnw eÐnai paradeÐgmata sunart sewn poll¸n metablht¸n.

An perioristoÔme se sunart seic dÔo metablht¸n tìte mporoÔme na sum-bolÐsoume mÐa tètoia sunrthsh me

z = f(x, y), (2.1.1)

ìpou x, y eÐnai oi anexrthtec metablhtèc. H grafik parstas thc s-ton q¸ro wc prìc èna kartesianì sÔsthma axìnwn Oxyz, eÐnai genik mÐaepifneia pou tèmnetai, apì mÐa eujeÐa parllhlh proc ton xona Oz, katto polÔ èna shmeÐo (monìtimh sunrthsh). MÐa sunrthsh tri¸n metablht¸nsumbolÐzetai me w = g(x, y, z), all den èqoume, sthn perÐptwsh aut , ènaeÔkolo trìpo gia thn grafik thc apeikìnish ston q¸ro.

Onomzoume isostajmikèc kampÔlec mÐac sunrthshc dÔo metablht¸n tickampÔlec pnw sto epÐpedo Oxy pou prokÔptoun apì tom thc epifneiac methn oikogèneia twn orizìntiwn epipèdwn z = k, k ∈ R. Ja èqoume loipìn thnmonoparametrik oikogèneia kampul¸n f(x, y) = k, k ∈ R kai sqedizontacorismènec apì autèc sto epÐpedo Oxy mporoÔme na èqoume kalÔterh epopteÐagia thn sumperifor thc sunrthshc ston q¸ro.

EpÐshc orÐzoume wc x−profÐl thc f , thn oikogèneia twn kampul¸n pnwsto epÐpedo Oyz pou prokÔptoun wc tom thc f me ta epÐpeda x = x0. Araja eÐnai z = f(x0, y), x0 ∈ R. Parìmoia orÐzetai to y−profÐl.

Pardeigma 2.1: Estw to elleiptikì paraboloeidèc z = x2 + y2. Oiisostajmikèc kampÔlec x2 + y2 = k ≥ 0 eÐnai kÔkloi, en¸ oi kampÔlec pousunistoÔn to x−profÐl thc sunrthshc, pnw sto epÐpedo Oyz, dÐnontai apìthn z = y2 + x20, x0 ∈ R. Ed¸ èqoume parabolèc.

2.2 Tìpoi, PedÐa OrismoÔ


2.2. TOPOI, PEDIA ORISMOU 21

Epeid to zeÔgoc twn anexrthtwn metablht¸n (x, y), se mÐa sunrthshdÔo metablht¸n, genik an kei sto epÐpedo R2, eÐnai qr simo na exetsoumesunoptik kpoiec idiìthtec twn uposunìlwn tou. Kat' arq n h apìstashdÔo shmeÐwn sto epÐpedo eÐnai

d(P, P′) =

√(x− x′)2 + (y − y′)2 (2.2.1)

Onomzoume δ−perioq enìc shmeÐou P0 to sÔnoloΠ(P0, δ) = {P (x, y)/d(P, P0) < δ} (2.2.2)

AntistoÐqwc tetragwnik d-perioq onomzoume to sÔnolo

Π�(P0, δ) = {(x, y)/|x− x0| < δ, |y − y0| < δ} (2.2.3)Anoiktì sÔnolo D ⊆ R2 onomzoume to sÔnolo pou gia kje shmeÐo touP0 uprqei perioq Π(P0, δ) pou na perièqetai ex' olokl rou s' autì, dhl.Π(P0, δ) ⊂ D.Sumpl rwma enìc sunìlou onomzoume to C(D) = R2 −D.Kleistì onomzoume èna sÔnolo ìtan to sumpl rwm tou eÐnai anoiktì sÔno-lo.Ena sÔnolo D onomzetai sunafèc an gi kje zeÔgoc shmeÐwn tou P1, P2uprqei tejlasmènh gramm me peperasmèno pl joc tmhmtwn pou ta en¸neikai an kei ex' olokl rou sto D.Tìpoc onomzetai èna sunafèc uposÔnolo tou epipèdou. Oi tìpoi me toucopoÐouc ja asqolhjoÔme kurÐwc ed¸, mporoÔn na grafoÔn se mÐa apì tic dÔoparaktw morfèc

D1 = {(x, y)/a ≤ x ≤ b, f1(x) ≤ y ≤ f2(x)}D2 = {(x, y)/g1(y) ≤ x ≤ g2(y), c ≤ y ≤ d} (2.2.4)

Enac tìpoc ja lègetai fragmènoc an uprqei stajer M , ¸ste gia kjezeÔgoc shmeÐwn tou na èqoume d(P1, P2) < M .

Ena shmeÐo P0 ja onomzetai sunoriakì shmeÐo tou D, an kje d-perioq tou shmeÐou autoÔ perièqei shmeÐa pou an koun ston tìpo, kaj¸c kai shmeÐ-a pou dèn an koun s' autìn. Tèloc to shmeÐo P0 ja onomzetai shmeÐosuss¸reushc tou tìpou D, an kje d-perioq tou perièqei shmeÐa pou an k-oun ston tìpo.

To pedÐo orismoÔ thc sunrthshc thc Ex. 2.1.1 eÐnai sun jwc enac tìpocpou sumbolÐzetai me D[f ]. Genik autìc prokÔptei Ôstera apì epÐlush kpoi-wn periorism¸n pou epibllei h morf thc sunrthshc. Oi kuriìteroi faÐnon-tai ston PÐnaka 2.1.

Pardeigma 2.2: An f1(x, y) = ln(1−x2− y2) ìpou me ln sumbolÐzoumeton nepèreio logrijmo, tìte D[f1] = {(x, y)/x2 + y2 < 1}, dhl. to pedÐoorismoÔ thc sunrthshc eÐnai to eswterikì tou monadiaÐou kÔklou.



Morf Periorismìc

logF (x, y) F (x, y) > 0

1/F (x, y) F (x, y) 6= 0

2n√

F (x, y) F (x, y) ≥ 0

Arcsin(F (x, y)) |F (x, y)| ≤ 1

PÐnakac 2.1: PeriorismoÐ sto pedÐo orismoÔ.

An f2(x, y) = 1/(1−x2−y2) tìte to D[f2] tautÐzetai me ìlo to epÐpedo ektìctwn shmeÐwn tou monadiaÐou kÔklou.Tèloc, ìtan èqoume polloÔc periorismoÔc, wc pedÐo orismoÔ prokÔptei o tìpocpou touc ikanopoieÐ tautìqrona ìlouc. An p.q. f3(x, y) = xln(y − 1) + (1−x2)1/2 tìte D[f3] = {(x, y)/y > 1, &−1 ≤ x ≤ 1} dhl. h lwrÐda tou epipèdouanmesa stic eujeÐec x = ±1 kai pnw apì thn eujeÐa y = 1.


2.3. ORIA 23

2.3 'Oria

Ja lème ìti mÐa sunrthsh z = f(x, y) èqei ìrio ton arijmì k, kaj¸cto shmeÐo P (x, y) plhsizei èna sugkekrimèno shmeÐo P0(x0, y0) kai ja tosumbolÐzoume wc

limP→P0

f(x, y) = k (2.3.1)

an gia kje arijmì ² > 0 (osod pote mikrì) mporoÔme na broÔme èna arijmìδ = δ(P0, ²) > 0 tètoion ¸ste gia ìla ta shmeÐa P (x, y) pou ikanopoioÔnthn d(P, P0) < δ, na èqoume |f(x, y) − k| < ². O orismìc autìc mac lèei ìtiprèpei, kaj¸c plhsizoume to shmeÐo P0 (anexart twc diadrom c!), oi timècthc sunrthshc na proseggÐzoun to k. O orismìc autìc bohjei merikèc forèckai se apodeÐxeic orÐwn.

Pardeigma 2.3: DeÐxte ìti limP→P0x2y2

x2+y2= 0 ìpou P0(0, 0).

Dedomènou kpoiou ² > 0 prèpei na prosdiorÐsoume èna δ > 0 ¸ste ìtand(P, P0) =

√x2 + y2 < δ na isqÔei

|f(x, y)− k| = | x2y2

x2 + y2− 0| < ²

All |x| ≤√

x2 + y2 opìte x2 ≤ (x2 + y2) kai y2 ≤ (x2 + y2). Pollaplasi-zontac kat mèlh prokÔptei ìti |f(x, y) − k| ≤ (x2 + y2) < δ2 kai sunep¸cèqoume thn zhtoÔmenh anisìthta, an epilèxoume δ = δ(²) =

√².

Je¸rhma: To ìrio en uprqei eÐnai monadikì.Apìdeixh: En limP→P0 f(x, y) = k1 kai limP→P0 f(x, y) = k2 tìte gia kje² > 0 uprqoun δ1, δ2 > 0 ¸ste ìtan d(P, P0) < δ1 (antÐstoiqa d(P, P0) < δ2)na èqoume |f − k1| < ² (antÐstoiqa |f − k2| < ²). Ara n epilèxoume δ =min(δ1, δ2) tìte

|k1 − k2| = |f − k2 − (f − k1)| ≤ |f − k1|+ |f − k2| < 2²

ìtan d(P, P0) < δ kai lìgw tou ìti to ² mporeÐ na gÐnei aujaÐreta mikrì jaèqoume k1 = k2 . ¤Qr simh praktik sunèpeia tou jewr matoc eÐnai ìti an gia dÔo diaforetikoÔcdrìmouc prosèggishc tou shmeÐou P0 prokÔptoun diaforetikèc oriakèc timèc,tìte to ìrio den uprqei. IsodÔnamoc eÐnai kai o orismìc pou knei qr sh miacakoloujÐac shmeÐwn sto R2.

Orismìc: An to P0 eÐnai shmeÐo suss¸reushc tou D[f ], tìte h Ex.(2.3.1) eÐnai isodÔnamh me to akìloujo gegonìc: Gia kje akoloujÐa shmeÐ-wn {Pn, n = 1, 2, ...} pou sugklÐnei sto P0 kai an kei sto pedÐo orismoÔ, hakoloujÐa {f(Pn)} twn tim¸n thc sunrthshc sugklÐnei sto k.

Pardeigma 2.4: DeÐxte ìti to lim(x,y)→(0,0) xyx2+y2 dèn uprqei.'Estw h akoloujÐa shmeÐwn Pn(xn, yn) tètoia ¸ste yn = mxn, m ∈ R



dhl. proseggÐzoume thn arq mèsw eujei¸n pou dièrqontai apì aut n. Tìtef(x, y) = m/(1 + m2) kai sunep¸c to ìrio den uprqei giatÐ h tim touexarttai apì ton diadrom prosèggishc thc arq c (dhl. apì to m).

Pardeigma 2.5: DeÐxte ìti to ìrio thc f(x, y) = xy2/(x2 + y4) sthnarq , den uprqei.Kat m koc thc eujeÐac y = x èqoume

lim(x,y)→(0,0)

f(x, y) = limx→0

x3

x2 + x4= 0

en¸ kat m koc thc kampÔlhc x = y2 èqoume

lim(x,y)→(0,0)

f(x, y) = limy→0

y4

y4 + y4=

1

2

Ara to ìrio den uprqei.Orismìc: 'Orio sto peiro. Ja lème ìti lim(x,y)→(∞,∞) f(x, y) = k an

gia kje ² > 0 uprqei δ = δ(²) > 0 ¸ste ìtan√

x2 + y2 > δ na èqoume|f(x, y)− k| < ².

Pardeigma 2.6: An f(x, y) = 1 + 1x2+y2

deÐxte ìti lim(x,y)→(∞,∞) = 1.Prgmati gia na èqoume |f(x, y) − 1| = 1/(x2 + y2) < ² arkeÐ na epilèxoumeδ(²) = 1/

√².

Sun jwc gia na broÔme to ìrio mÐac sunrthshc sto peiro, meletme toisodÔnamo ìrio

lim(x,y)→(∞,∞)

f(x, y) = lim(w,z)→(0,0)

f(1

w,1

z) (2.3.2)

Pardeigma 2.7: To paraktw ìrio

lim(x,y)→(∞,∞)

x2 + y2 + 2

x2 + y2= lim

(w,z)→(0,0)1/w2 + 1/z2 + 2

1/w2 + 1/z2=

= 1 + 2 lim(w,z)→(0,0)

w2z2

w2 + z2= 1 + 2 · 0 = 1

Diadoqik 'Oria. Ta diadoqik ìria anafèrontai se sugkekrimènec di-adromèc prosèggishc tou oriakoÔ shmeÐou P0, dhl. pr¸ta parllhla wc prìcèna xona kai katìpin wc prìc ton llo. OrÐzoume

k1 = limy→y0

[ limx→x0

f(x, y)]

k2 = limx→x0

[ limy→y0

f(x, y)] (2.3.3)

IsqÔoun genik ta akìlouja:


2.4. SUNEQEIA, OMOIOMORFH SUNEQEIA 25

1. An uprqei to k kai uprqoun epÐshc kai ta k1, k2 tìte k = k1 = k2('Askhsh 7).

2. Apo mìno tou to k den eggutai thn Ôparxh twn k1, k2 ('Askhsh 8).

3. 'Otan ta diadoqik ìria uprqoun kai eÐnai Ðsa (k1 = k2) den shmaÐneiìti aparaÐthta uprqei kai to ìrio k ('Askhsh 9).

4. 'Otan k1 6= k2 tìte to k den uprqei ('Askhsh 10).5. MporeÐ na uprqei èna apì ta k1, k2 kai to k, qwrÐc na uprqei to llo

diadoqikì ìrio ('Askhsh 11).

To ìrio tou ajroÐsmatoc, diaforc, ginomènou kai phlÐkou dÔo sunart -sewn kaj¸c kai ta ìria twn rÐzwn, akoloujoÔn se genikèc grammèc touc Ðdiouckanìnec me aut twn sunart sewn mÐac metablht c. Ed¸ ja kleÐsoume thnenìthta me kpoiec piì endiafèrousec idiìthtec

I1. An isqÔoun |f(x, y) − k| ≤ |g(x, y)| kai limP→P0 g(x, y) = 0 tìte jaèqoume kai limP→P0 f(x, y) = k.

I2. To ginìmeno mhdenik c sunrthshc epÐ fragmènh eÐnai mhdenik . Anf(x, y) = g(x, y)h(x, y), |g(x, y)| ≤ M (fragmènh) kai limP→P0 h(x, y) = 0tìte ja eÐnai kai limP→P0 f(x, y) = 0 ('Askhsh 12).

I3. An f(x, y) ≤ g(x, y) ≤ h(x, y) kai isqÔoun tautìqrona ìti limP→P0 f(x, y) =k kai limP→P0 h(x, y) = k tìte ja eÐnai kai limP→P0 g(x, y) = k ('Askhsh 13).

2.4 Sunèqeia, Omoiìmorfh Sunèqeia

MÐa sunrthsh eÐnai suneq c se èna shmeÐo tou pedÐou orismoÔ tou an toìrio thc sto shmeÐo autì sumpÐptei me thn arijmhtik thc tim , dhl.

limP→P0

f(x, y) = k = f(x0, y0) (2.4.1)

IsqÔoun ta paraktw:

A. An mÐa sunrthsh eÐnai suneq c sto P0 tìte uprqei perioq tou shmeÐouautoÔ sthn opoÐa h f eÐnai fragmènh.

B. An h sunrthsh eÐnai suneq c se kleistì kai fragmèno tìpo tìte eÐnaifragmènh.

An mÐa sunrthsh den eÐnai suneq c se èna shmeÐo tìte lègetai asuneq c. Oipijanèc peript¸seic eÐnai: 1. na m n uprqei to ìrio, 2. h arijmhtik tim thcsunrthshc na eÐnai diaforetik tou orÐou kai 3. na mhn orÐzetai h sunrthshsto shmeÐo autì.



Pardeigma 2.8: BreÐte to λ ∈ R ¸ste h sunrthsh

f(x, y) =

{(1 + x2)

(siny

y

)(x, y) 6= (0, 0)

λ (x, y) = (0, 0)

na eÐnai suneq c. EÔkola brÐskoume ìti lim(x,y)→(0,0) f(x, y) = 1 = λ.An mÐa sunrthsh eÐnai suneq c se kje shmeÐo enìc tìpou tìte lème ìti

eÐnai suneq c ston tìpo autìn. H ènnoia thc sunèqeiac eÐnai topik ènnoia,dhl. anafèretai sun jwc se èna shmeÐo. H ènnoia thc omoiìmorfhc sunèqeiacèqei kajolikì qarakt ra sta plaÐsia enìc tìpou.

An P0 tuqaÐo shmeÐo tou D[f ] tìte genik ston orismì thc sunèqeiacèqoume |f(P ) − f(P0)| < ² en epilèxoume d(P, P0) < δ ìpou genik toδ = δ(P0, ²). ParathroÔme ìti o ”rujmìc” prosèggishc thc oriak c tim cexarttai apì to shmeÐo P0. An t¸ra isqÔei δ = δ(²), anexart twc tou shmeÐouP0 lème ìti h sunrthsh eÐnai omoiìmorfa suneq c ston tìpo D.

IsodÔnamoc eÐnai kai o orismìc:H f(x, y) eÐnai omoiìmorfa suneq c ston D ⇐⇒ ∀² > 0 uprqei δ = δ(²) > 0¸ste ∀P1, P2 ∈ D me d(P1, P2) < δ na èqoume |f(P1)− f(P2)| < ².

2.5 Ask seic

1. UpologÐste tic timèc twn paraktw sunart sewn sta ex c shmeÐa: f1(x, y) =2x2y + x + y3 sta (1, 0), (−2, 1), (1, 1

2) kai f2(x, y) = xlny + e−xy sta

(1, 1), (0, 12), (−1, 2). DeÐxte ìti h sunrthsh

z =x4 + 2x2y2 + y4

1− x2 − y2

èqei stajer tim pnw ston kÔklo aktÐnac R. UpologÐste thn.

2. BreÐte tic isostajmikèc kampÔlec kai tic kampÔlec tou x, y−profÐl giathn z = 2x + 3y. To Ðdio gia to elleiyoeidèc, ton k¸no, to elleiptikìkai uperbolikì paraboloeidèc, ìpwc aut dÐnontai sto prohgoÔmeno ke-flaio.

3. Na brejoÔn kai na sqediastoÔn ta pedÐa orismoÔ twn paraktw sunart -sewn z = f(x, y):

z =1√

x− y +1√

x + y+ 2

z =√

cos(x2 + y2)

z = 3y2 − 9x + 5ln(x2)


2.5. ASKHSEIS 27

z = x/(y2 − 4x)z =

√x2 + y2 − 4 ln(16− x2 − y2)

z = (x2 − y2)1/2 + (x2 − y2 − 1)1/2z = ln(x + y)

z = ln(x2 + y)

z = Arccos(xy)

4. DeÐxte me bsh ton orismì ìti lim(x,y)→(1,2)[x2 − 2x + y] = 1, kai epÐshclim(x,y)→(0,0) f(x, y) = 0 gia tic f1(x, y) = (x3 + y3)/(x2 + y2),f2(x, y) = (2x

3 − y3)/(x2 + y2) kai f3(x, y) = (x4 + y4)/(x2 + y2).5. DeÐxte ìti den uprqoun ta ìria sthn arq O(0, 0) gia tic paraktw

sunart seic, deÐqnontac ìti exart¸ntai apì thn diadrom .

f1 =xy + y3

x2 + y2, f2 =

x2y2

x2y2 + (x− y)2 ,

f3 =xy3

x2 + y6, f2 =

x2y4

(x2 + y4)3,

6. DeÐxte me bsh ton orismì ìti

lim(x,y)→(∞,∞)

sin(x2 + y2)

x2 + y2= 0

qrhsimopoi¸ntac to gegonìc ìti sin(u) ' u − u3/6 kai me katllhlhantikatstash ìti

lim(x,y)→(∞,∞)

2(x2 + y2)

x2 + y2 + 1= 2

7. DeÐxte ìti k = k1 = k2 gia thn f(x, y) = 2x + 3y kai P0(1, 2).

8. DeÐxte me bsh ton orismì ìti k = 0 gia thn f(x, y) = xsin(1/y) +ysin(1/x) sto P0(0, 0) en¸ kanèna apì ta diadoqik ìria den uprqeilìgw thc anuparxÐac tou orÐou limx→0 sin(1/x)!.

9. DeÐxte ìti an f(x, y) = xy/(x2 + y2) tìte k1 = k2 = 0 sto shmeÐoP0(0, 0) all to ìrio k den uprqei.

10. Gia thn

f(x, y) =x− y + x2 + y2

x + y

deÐxte ìti k1 = −1, k2 = 1 kai ra to ìrio k den uprqei.



11. Gia thn f(x, y) = xsin(1/y) deÐxte me bsh ton orismì ìti k = 0, k1 = 0,all ìti to k2 den uprqei.

12. DeÐxte ìti

lim(x,y)→(1,2)

sin(x2 + y2)√

(x− 1)2 + (y − 2)2 = 0.

13. An 1 ≤ f(x, y) ≤ sin(x + y)/(x + y) deÐxte ìti lim(x,y)→(0,0) f(x, y) = 1.14. Melet ste thn sunèqeia twn sunart sewn

f1(x, y) =

{ 3xysinxsiny

(x, y) 6= (0, 0)2 (x, y) = (0, 0)

f2(x, y) =

{x3y−xy3x2+y2

(x, y) 6= (0, 0)0 (x, y) = (0, 0)

15. OmoÐwc thn sunèqeia thc

f(x, y) =

{(1 + y2) sinxcosx

2x(x, y) 6= (0, 0)

λ (x, y) = (0, 0)

16. OmoÐwc thn sunèqeia thc

f(x, y) =

{ xyx2+xy+y2

(x, y) 6= (0, 0)0 (x, y) = (0, 0)

17. Melet ste thn omoiìmorfh sunèqeia twn paraktw sunart sewn z =f(x, y) sto epÐpedo R2: z = x2 + y2, z = x + 2y + y2, z = x2 + xy,z = x + 2y, z = x + y2.

18. DeÐxte ìti

lim(x,y)→(1,1)

x3 − x2 + x− 2xy + 2y − 1x− 1 = 0


Keflaio 3

Parag¸gish

29

30 KEFALAIO 3. PARAGWGISH

3.1 Merikèc Pargwgoi

H ènnoia thc parag¸gou, ìpwc thn gnwrÐzoume apì tic sunart seic mÐacmetablht c, genikeÔetai kai gia sunart seic dÔo perissotèrwn metablht¸nmèsw thc ènnoiac thc merik c parag ģou. An P0(x0, y0) eÐnai shmeÐo tou pedÐouorismoÔ thc sunrthshc f(x, y) tìte oi merikèc pargwgoi thc f wc prìc x, ysto shmeÐo autì, orÐzontai wc oi arijmoÐ

f′x(P0) = f

′x(x0, y0) = lim

∆x→0f(x0 + ∆x, y0)− f(x0, y0)

∆x

f′y(P0) = f

′y(x0, y0) = lim

∆y→0f(x0, y0 + ∆y)− f(x0, y0)

∆y(3.1.1)

Dhlad kat thn parag¸gish wc prìc mÐa metablht , kratme stajer thn llh ( tic llec). H merik pargwgoc f ′x(P0) mac dÐnei thn klÐsh thcefaptìmenhc sto shmeÐo P0, thc kampÔlhc pou prokÔptei wc tom thc z =f(x, y) me to epÐpedo y = y0. Anlogo sumpèrasma isqÔei kai gia thn f

′y(P0)

se sqèsh me to epÐpedo x = x0. An orÐzontai oi pargwgoi se kje shmeÐoenìc tìpou D, tìte èqoume tic sunart seic twn merik¸n parag¸gwn pr¸thctxhc pou sumbolÐzontai kai wc

f′x(x, y) =

∂f(x, y)

∂x

f′y(x, y) =

∂f(x, y)

∂y(3.1.2)

Oi an¸terhc txhc merikèc pargwgoi lambnontai ap' eujeÐac me diadoqikècparagwgÐseic. Etsi oi deÔterhc txhc pargwgoi eÐnai oi fxx = (∂2f/∂x2),fyy = (∂

2f/∂y2) kaj¸c kai oi miktèc fxy = (∂2f/∂y∂x) kai fyx = (∂2f/∂x∂y).IsqÔei to akìloujo

Je¸rhma (Schwarz): An se kpoio tìpo D uprqoun kai eÐnai suneqeÐcoi fx, fy, fyx tìte to Ðdio isqÔei kai gia thn fxy kai èqoume

∂2f

∂x∂y=

∂2f

∂y∂x(3.1.3)

Pardeigma 2.1: Me bsh ton orismì breÐte tic parag¸gouc pr¸thc txhcthc f(x, y) = sin(xy) sto shmeÐo P0(0, π/2). Ja èqoume

f′x(P0) = lim

∆x→0f(∆x, π/2)− f(0, π/2)

∆x= lim

∆x→0sin(π

2∆x)

∆x=

π

2

f′y(P0) = lim

∆y→0f(0, π/2 + ∆y)− f(0, π/2)

∆y= lim

∆→00

∆y= 0

Pardeigma 2.2: BreÐte tic parag¸gouc pr¸thc kai deÔterhc txhc giathn f(x, y) = e−2xsiny. Ja èqoume (∂f/∂x) = −2e−2xsiny kai (∂f/∂y) =


3.1. MERIKES PARAGWGOI 31

e−2xcosy gia tic parag¸gouc pr¸thc txhc en¸ (∂2f/∂x2) = 4e−2xsiny,(∂2f/∂x∂y) = −2e−2xcosy kai (∂2f/∂y2) = −e−2xsiny gia tic parag¸goucdeÔterhc txhc.

Entel¸c anloga orÐzontai kai oi pargwgoi gia sunart seic tri¸n perissìterwn metablht¸n. Oi sunhjismènoi kanìnec parag¸gishc pou isqÔ-oun stic sunart seic mÐac metablht c èqoun efarmog kai ed¸. O plèonshmantikìc pou mac dieukolÔnei stic prxeic eÐnai o kanìnac parag¸gishcsÔnjethc sunrthshc. An f(x, y) = g[h(x, y)] tìte

∂f

∂x=

(dg(u)

du

)

u=h(x,y)

∂h

∂x(3.1.4)

kai omoÐwc wc prìc y.Pardeigma 2.3: An f(x, y) = ln(x2 + y2) tìte (∂f/∂x) = 2x/(x2 + y2) kaiomoÐwc wc prìc y.

Gia sunart seic mÐac metablht c h Ôparxh thc parag¸gou f ′(x0) se ènashmeÐo tou pedÐou orismoÔ touc exasfalÐzei kai thn sunèqeia thc sunrthshcsto shmeÐo autì. Autì paÔei na isqÔei genik gia tic sunart seic dÔo metabl-ht¸n. To akìloujo digramma parousizei thn sqèsh sunèqeiac kai parag-wgisimìthtac gia sunart seic dÔo metablht¸n,

'Uparxh merik¸n parag¸gwn f ′x(P0), f′y(P0)

Suneq c sto P0

DiaforÐsimh sto P0

C1 sto P0

1.

2.3.4.

Sq ma 3.1: Sqèsh sunèqeiac kai paragwgisimìthtac

ìpou oi ènnoiec thc diaforisimìthtac kai thc C1−diaforisimìthtac jaoristoÔn amèswc paraktw. San pardeigma tou stoiqeÐou 1. sto Sq ma3.1, dhl. sunrthshc gia thn opoÐa uprqoun oi pr¸tec merikèc pargwgoiqwrÐc na eÐnai suneq c se èna shmeÐo, jewroÔme thn sunrthsh

f1(x, y) =

{ xyx2+y2

(x, y) 6= (0, 0)0 (x, y) = (0, 0)

(3.1.5)



Lìgw thc anuparxÐac tou orÐou sthn arq , h sunrthsh den eÐnai suneq call mporeÐte ne deÐxete ìti oi merikèc thc pargwgoi wc proc x, y upr-qoun ('Askhsh 4). 'Ena pardeigma tou stoiqeÐou 2. eÐnai h sunrthsh thc'Askhshc 5, ìpou mporeÐte na deÐxete ìti en¸ eÐnai suneq c, stereÐtai merik¸nparag¸gwn pr¸thc txhc.

Gia thn katanìhsh thc ènnoiac thc diaforisimìthtac jewroÔme thn sunrthshmÐac metablht c f(x). H Ôparxh thc parag¸gou sto shmeÐo x0 tou pedÐouorismoÔ thc se sundiasmì me to anptugma Taylor deÔterhc txhc f(x) =f(x0)+f

′(x0)(x−x0)+(1/2)f ′′(x0)(x−x0)2 eggu¸ntai ìti h eujeÐa Y−f(x0) =

f′(x0)(x − x0) efptetai thc sunrthshc sto shmeÐo P0(x0, f(x0)). Autì di-

asfalÐzetai ìtan

limx→x0

Y − f(x)|x− x0|

Aut h apaÐthsh genikeÔetai kai stic sunart seic dÔo metablht¸n. Ja lèmeìti h sunrthsh z = f(P ) = f(x, y) eÐnai diaforÐsimh sto shmeÐo P0(x0, y0)ìtan isqÔei

limP→P0

f(P )− Z(P )|P − P0| (3.1.6)

ìpou to epÐpedo Z(P ) = f(P0) + f′x(P0)(x − x0) + f ′y(y − y0) ja eÐnai tìte

efaptìmeno thc sunrthshc sto shmeÐo P0 kai |P−P0| = d(P, P0). H sunj khaut eÐnai isodÔnamh kai me thn Ôparxh twn parag¸gwn deÔterhc txhc thcf(x, y).

Pardeigma 2.4: An z = f(x, y) = x2 + y2 kai P0(1, 1) tìte to epÐpedoìpwc dÐnetai apì thn parapnw sqèsh eÐnai to Z = 2x + 2y − 2 kai eÐnaiprgmati efaptìmeno thc f diìti

limP→P0

f(P )− Z(P )|P − P0| = lim(x,y)→(1,1)

x2 + y2 − 2x− 2y + 2√(x− 1)2 + (y − 1)2 = 0

Tèloc mÐa sunrthsh ja eÐnai C1−diaforÐsimh en oi pr¸tec merikèc thcpargwgoi (wc sunart seic plèon) uprqoun kai eÐnai suneqeÐc ston tìpoD. 'Ena pardeigma sunrthshc ìpwc to stoiqeÐo 3. pou eÐnai suneq c allìqi diaforÐsimh dÐnetai sthn 'Askhsh 6, en¸ èna pardeigma sunrthshc ìp-wc to stoiqeÐo 4. pou eÐnai diaforÐsimh all ìqi C1 dÐnetai sthn 'Askhsh 7.Suqnìtera ja asqolhjoÔme me sunart seic C1 pou plhroÔn ìlec tic ”kalèc”sunj kec omalìthtac kai sunèqeiac wc prìc thn diaforisimìthta.

3.2 Armonikèc kai OmogeneÐc Sunart seic


3.3. SUNJETES KAI PEPLEGMENES SUNARTHSEIS 33

MÐa sunrthsh f(x, y) lègetai armonik an ikanopoieÐ thn exÐswsh Laplace

∂2f

∂x2+

∂2f

∂y2= 0 (3.2.1)

AntÐstoiqa stic treÐc diastseic h sunrthsh w = F (x, y, z) ja lègetai ar-monik an ikanopoieÐ thn

Fxx + Fyy + Fzz = 0 (3.2.2)

Gnwst armonik sunrthsh stic treÐc diastseic eÐnai h sunrthsh tou h-lektrikoÔ dunamikoÔ apousÐa fortÐwn, φ = 1/

√x2 + y2 + z2 (apodeÐxte to).

Pardeigma 2.5: Gia thn z = f(x, y) = ln(x2 + y2) èqoume ìti fxx =(−2x2 + 2y2)/(x2 + y2)2 kai ìmoia sqèsh gia thn fyy ìpote diapist¸noume ìtieÐnai armonik sunrthsh stic dÔo diastseic.

MÐa sunrthsh ja lègetai omogen c bajmoÔ m en isqÔei

f(λx, λy) = λmf(x, y) (3.2.3)

ìpou m akèraioc. IsqÔei to akìloujoJe¸rhma (Euler): Gia mÐa omogen sunrthsh bajmoÔ m

x∂f

∂x+ y

∂f

∂y= mf (3.2.4)

Pardeigma 2.6: H z = x2 + xy + y2 eÐnai omogen c deutèrou bajmoÔdiìti f(λx, λy) = (λx)2 + (λx)(λy) + (λy)2 = λ2f(x, y).

3.3 SÔnjetec kai Peplegmènec Sunart seic

'Estw z = f(x, y) mÐa sunrthsh kai x = g(r, s), y = h(r, s) ènac metasqh-matismìc pou sthn ousÐa mac phgaÐnei apì to zeÔgoc twn palai¸n metablht¸n(x, y) stic kainoÔrgiec (r, s). Tìte ja èqoume genik z = f [g(r, s), h(r, s)] =F (r, s) kai n jèloume tic merikèc parag¸gouc thc f wc prìc tic kainoÔrgiecmetablhtèc tìte zhtme na paragwgÐsoume thn prokÔptousa sÔnjeth sunrthsh.Ja eÐnai tìte

∂z

∂r=

∂f

∂x

∂g

∂r+

∂f

∂y

∂h

∂r∂z

∂s=

∂f

∂x

∂g

∂s+

∂f

∂y

∂h

∂s(3.3.1)

AfoÔ upologÐsoume ìlec tic merikèc parag¸gouc, prèpei telik to apotèlesmana to ekfrsoume wc prìc (r, s).



Pardeigma 2.7: An z = Arctan(x/y) kai x = rsins, y = coss tìte

∂z

∂r=

(1/y)

1 + (x/y)2sins− x/y

2

1 + (x/y)2· 0 = sinscoss

r2sin2s + cos2s

∂z

∂s=

(1/y)

1 + (x/y)2rcoss +

x/y2

1 + (x/y)2sins =

r

r2sin2s + cos2s

An t¸ra x = g(t), y = h(t) dhl. èqoume metbash se mÐa metablht , tìte holik pargwgoc

dz

dt=

∂f

∂x

dg

dt+

∂f

∂y

dh

dt(3.3.2)

Pardeigma 2.8: An z = (x/y) kai x = et, y = lnt, tìte

dz/dt = (1/y)et − (x/y2)(1/t) = et

lnt− e

t

t(lnt)2

Upojètoume t¸ra ìti èqoume mÐa sqèsh thc morf c F (x, y) = 0. Aut orÐzei upì proupojèseic sunrthsh mÐac metablht c y = f(x) akìma kai anden mporeÐ na epilujeÐ rht (ekpefrasmèna) wc prìc y. Tìte lème ìti èqoumemÐa peplegmènh sunrthsh. H proupìjesh gia na ekfrzei h F (x, y) = 0 mÐapeplegmènh sunrthsh, sthn geitoni enìc shmeÐou P0, eÐnai F

′y(P0) 6= 0.

MporoÔme t¸ra na lboume thn pr¸th kai deÔterh pargwgo thc sunrthshcy = f(x) wc ('Askhsh 12)

y′(x) = −Fx

Fy

y′′(x) = −(FxxF

2y − 2FxyFxFy + FyyF 2x )

F 3y(3.3.3)

ìpou efex c paraleÐpoume ton tìno pnw apì tic merikèc parag¸gouc giaaplopoÐhsh tou sumbolismoÔ kai jewroÔme ìti sto dexiì mèloc twn sqèsewnaut¸n to y ekfrzetai wc prìc to x mèsw thc F (x, y) = 0.

Pardeigma 2.9: 'Estw F (x, y) = xey − yex = 0. EÐnai fanerì ìtiaut h sqèsh den mporeÐ na epilujeÐ rht wc prìc y. All ja èqoume y′ =(ey − yex)/(ex − xey). UpologÐste san skhsh thn deÔterh pargwgo.

OmoÐwc an èqoume mÐa sqèsh F (x, y, z) = 0 aut ja orÐzei mÐa pepleg-mènh sunrthsh z = z(x, y) en isqÔei F (x, y, z(x, y)) = 0. Gia na isqÔeiautì, dhl. akribèstera gia na mporoÔme na poÔme ìti orÐzetai mÐa sunrthshdÔo metablht¸n upì peplegmènh morf sthn perioq Π(P0(x0, y0, z0), δ) toushmeÐou P0 ja prèpei oi F, Fz na eÐnai suneqeÐc sthn perioq aut kai naèqoume F (P0) = 0, Fz(P0) 6= 0.


3.4. IAKWBIANES, DIAFORIKA 35

3.4 Iakwbianèc, Diaforik

An èqoume metbash apì ena zeÔgoc metablht¸n (x, y) se èna llo (u, v),mèsw tou metasqhmatismoÔ u = u(x, y), v = v(x, y) tìte orÐzoume ton Iakw-bianì pÐnaka kai thn antÐstoiqh orÐzous tou wc

D(u, v)

D(x, y)=

∣∣∣∣ux, uyvx, vy

∣∣∣∣ (3.4.1)

'An h orÐzousa aut eÐnai diforh tou mhdenìc sthn perioq shmeÐou P0, tìteuprqei kai o antÐstrofoc metasqhmatismìc kai isqÔei

D(u, v)

D(x, y)· D(x, y)D(u, v)

= 1 (3.4.2)

MÐa basik idiìthta twn iakwbian¸n orizous¸n eÐnai ìti sundèoun ta stoiqei¸d-h embad stic antÐstoiqec suntetagmènec, dhl.

dudv =D(u, v)

D(x, y)dxdy (3.4.3)

Pardeigma 2.10: An èqoume x = rcosθ, y = rsinθ tìte paÐrnoumethn gnwst sqèsh pou sundèei ta stoiqei¸dh embad twn kartesian¸n me ticpolikèc suntetagmènec

dxdy =

∣∣∣∣cosθ, −rsinθsinθ, rcosθ

∣∣∣∣ drdθ = rdrdθ

AntÐstoiqec efarmogèc stic treÐc diastseic uprqoun stic ask seic sto tèloctou kefalaÐou.

Mia llh qr simh idiìthta thc iakwbian c eÐnai ìti ìtan aut mhdenÐze-tai tautotik se kpoio tìpo, tìte oi sunart seic u, v den eÐnai sthn prag-matikìthta anexrthtec all uprqei metaxÔ touc exrthsh (sunarthsiak sqèsh).

Pardeigma 2.11: 'Estw u = (x+y)/(1−xy), v = Arctanx+Arctany.Tìte D(u, v)/D(x, y) = 0 prgma pou shmaÐnei ìti den eÐnai anexrthtec oinèec metablhtèc. Pragmatik eÔkola brÐskoume ìti isqÔei h sunarthsiak sqèsh u = tanv.

Erqìmaste t¸ra na melet soume thn ènnoia tou diaforikoÔ. 'Estw ìti∆x = dx, ∆y = dy eÐnai mikrèc (apeirostèc) aux seic twn anexrthtwnmetablht¸n (x, y) gÔrw apì to shmeÐo P0(x0, y0). OrÐzoume to diaforikì thcsunrthshc f(x, y) wc

df = dz =

(∂f

∂x

)

P0

dx +

(∂f

∂y

)

P0

dy (3.4.4)



Pollèc forèc qrhsimopoioÔme to diaforikì gia proseggistikèc ektim seic twntim¸n mÐac sunrthshc. Prgmati an orÐsoume ∆f := f(x0 + ∆x, y0 + ∆y)−f(x0, y0) tìte lìgw kai tou Jewr matoc Taylor pou ja doÔme paraktw m-poroÔme na laboume proseggistik ∆f ' df kai ra na ektim soume tof(x0 + ∆x, y0 + ∆y).

Pardeigma 2.12: Ektim ste to I = [(1.01)2+(4.97)2+1]1/3. JewroÔmethn sunrthsh z = f(x, y) = [x2 + y2 + 1]1/3 kai epilègoume P0(1, 5) me∆x = 0.01, ∆y = −0.03. Tìte h zhtoÔmenh tim I = f(x0 + ∆x, y0 + ∆y) 'f(x0, y0) + df . All f(P0) = 3, (∂f/∂x)0 = 2/27 kai (∂f/∂y)0 = 10/27.'Ara ja eÐnai I ' 3 + (2/27)0.01 − (10/27)0.03 = 2.99. Gia sÔgkrish, me toepisthmonikì kompiouterki brÐskoume I = 2.9896.

MporoÔme t¸ra na jèsoume kai to antÐstrofo prìblhma, se ìti afor todiaforikì. An mac dojeÐ mÐa sqèsh thc morf c P (x, y)dx + Q(x, y)dy eÐnaidunatìn na paristnei to diaforikì kpoiac sunrthshc f(x, y) kai an naipoic; Ikan kai anagkaÐa sunj kh gi' autì eÐnai na isqÔei

∂P (x, y)

∂y=

∂Q(x, y)

∂x(3.4.5)

kai h zhtoÔmenh sunrthsh dÐnetai apì thn

f(x, y) =

∫ xx0

P (t, y)dt +

∫ yy0

Q(x0, u)du + C (3.4.6)

ìpou C aujaÐreth stajer.Pardeigma 2.13: 'Estw h parstash (excosy − eysinx)dx + (eycox −

exsiny)dy. Aut eÐnai to diaforikì mÐac sunrthshc diìti (∂P/∂y) = −exsiny−eysinx = (∂Q/∂x) kai efarmìzontac thn sqèsh thc Ex. (3.4.6) brÐskoume ìtif(x, y) = excosy + eycosx + C.

3.5 Polu¸numo Taylor

Me to diaforikì katafèrame na proseggÐsoume tic timèc thc sunrthshcse èna shmeÐo P (x, y) apì upologismoÔc se èna ”kontinì” shmeÐo P0(x0, y0).To polu¸numo Taylor epekteÐnei aut th mèjodo.

Je¸rhma (Taylor): An h f(x, y) èqei parag¸gouc mèqri txhc n suneqeÐcsthn perioq enìc shmeÐou P0(x0, y0) kai uprqei kai h pargwgoc txhc (n+1)sthn Ðdia perioq , tìte

f(x0 + h, y0 + k) = f(x0, y0) + (h∂x + k∂y)f(x0, y0) +

+1

2!(h∂x + k∂y)

2f(x0, y0) +


3.6. PARAGWGISH OLOKLHRWMATOS 37

+ ....................................... +

+1

n!(h∂x + k∂y)

nf(x0, y0) + Rn+1(3.5.1)

ìpou to upìloipo Rn+1 = 1(n+1)!(h∂x +k∂y)n+1f(x0+θh, y0+θk) upologÐzetai

se kpoio endimeso shmeÐo twn P0(x0, y0) kai P (x = x0 +h, y = y0 +k) opìte0 < |θ| < 1. Anloga me thn akrÐbeia pou epijumoÔme ston upologismì thctim c f(x, y) kratme ìrouc mèqri kpoiac txhc n. Sun jwc limn→∞ Rn+1 =0 opìte èqoume thn seir Taylor, en¸ an P0(x0, y0) = (0, 0) èqoume thn seirMac-Laurin.

Sta anaptÔgmata Taylor qrhsimopoioÔme to di¸numo tou NeÔtwna

(A + B)n =n∑

k=0

(nk

)AkBn−k,

(nk

)=

n!

k!(n− k)! (3.5.2)

Wc pardeigma, an jèloume na krat soume ìrouc mèqri deÔterhc txhc jaèqoume

f(x, y) = f(x0, y0) + (hfx(P0) + kfy(P0)) +

+1

2(h2fxx(P0) + 2hkfxy(P0) + k

2fyy(P0)) +

+ O(h3, k3) (3.5.3)ìpou h = (x−x0), k = (y−y0) kai o teleutaÐoc ìroc dhl¸nei ìti paraleÐpoumeìrouc trÐthc txhc wc prìc h, k. Autì gÐnetai eÐte diìti 0 < |h|, |k| < 1 eÐtediìti krÐnoume ìti eÐnai ikanopoihtik h akrÐbeia pou lambnoume me ìroucmèqri kai deÔterhc txhc.

Pardeigma 2.14: An f(x, y) = 1/√

1 + x + y kai P0(0, 0) breÐte topolu¸numo Mac-Laurin deÔterhc txhc gia thn sunrthsh aut . Epeid fx = fy = −1/2(1 + x + y)3/2 kai fxx = fxy = fyy = 3/4(1 + x + y)5/2antikajist¸ntac sthn parapnw sqèsh prokÔptei

f(x, y) =1√

1 + x + y= 1− 1

2(x + y) +

3

8(x2 + 2xy + y2) +

+ O(x3, y3)H prosèggish miac opoiasd pote sunrthshc me èna polu¸numo eÐnai polÔqr simh, diìti ta polu¸numa eÐnai gnwstèc sunart seic pou eÔkola mporoÔmena qeiristoÔme se prxeic. Autì eÐnai to pleonèkthma pou mac prosfèrei toanptugma Taylor.

3.6 Parag¸gish Oloklhr¸matoc



An mac dÐnetai h sunrthsh mÐac metablht c

F (x) =

∫ φ2(x)φ1(x)

f(x, t)dt (3.6.1)

dhl. upì thn morf oloklhr¸matoc, tìte mporoÔme na thn paragwgÐsoumesÔmfwna me ton kanìna Leibniz kai

F′(x) =

∫ φ2(x)φ1(x)

∂f(x, t)

∂xdt + f [x, φ2(x)]φ

′2(x)− f [x, φ1(x)]φ

′1(x) (3.6.2)

San eidik perÐptwsh, jètontac φ1(x) = 0, φ2(x) = x kai f(x, t) = f(t) lam-bnoume to jemelei¸dec je¸rhma olokl rwshc gia sunart seic mÐac metabl-ht cF′(x) = d

dx

∫ x0

f(t)dt = f(x).

Pardeigma 2.15: ParagwgÐste thn F (x) =∫ x2

xsin(xt)

tdt. To olokl rw-

ma autì den mporeÐ na upologisteÐ rht ¸ste na proume met thn pargwgìtou. Omwc efarmìzontac ton kanìna Leibniz èqoume

F′(x) =

∫ x2

x

∂

∂x

(sin(xt)

t

)dt +

sin(x3)

x2(x2)

′ − sin(x2)

x(x)

′=

=

∫ x2

x

cos(xt)dt +2sin(x3)

x− sin(x

2)

x=

=3sin(x3)− 2sin(x2)

x


3.7. AKROTATA, AKROTATA UPO SUNJHKH 39

3.7 Akrìtata, Akrìtata upì sunj kh

An h sunrthsh z = f(x, y) èqei topikì mègisto sto P0(x0, y0) tìte lìgwsunèqeiac uprqei perioq Π(P0, δ) ¸ste gia kje shmeÐo P ∈ Π(P0, δ) naèqoume f(P ) ≤ f(P0). Ja eÐnai f(x0 + h, y0 + k) − f(x0, y0) ≤ 0 ìpou eÐ-nai 0 < |h|, |k| < δ. 'Estw t¸ra ìti kinoÔmaste kat m koc tou y = y0.Knontac qr sh tou anaptÔgmatoc Taylor èqoume f(x0 + h, y0)− f(x0, y0) =fx(P0)h + (1/2)fxx(P0)h

2 < 0. Epeid jèloume h posìthta aut na eÐnai sta-jer arnhtik , anexart twc tou pros mou tou h upoqreoÔmaste na lboumefx(P0) = 0. Se anlogo sumpèrasma katal goume an kinhjoÔme kat m kocthc eujeÐac x = x0. ParathroÔme sunep¸c ìti h anagkaÐa sunj kh gia naèqoume akrìtato (mègisto elqisto) se èna shmeÐo P0 eÐnai

(∂f

∂x

)

P0

= 0 =

(∂f

∂y

)

P0

(3.7.1)

H sunj kh aut eÐnai anagkaÐa all ìqi kai ikan . Dhlad to sÔnolo twnlÔsewn thc (pou kaloÔntai krÐsima shmeÐa) perièqei kai shmeÐa pou dèn eÐnaiakrìtata. Gia na prosdiorÐsoume mÐa ikan sunj kh prèpei na krat soume kaiìrouc deÔterhc txhc sto anptugma Taylor. An P0 eÐnai èna krÐsimo shmeÐotìte

f(x0 + h, y0 + k)− f(x0, y0) = 12[Ah2 + 2Bhk + Ck2]

A :=

(∂2f

∂x2

)

P0

, B :=

(∂2f

∂x∂y

)

P0

C :=

(∂2f

∂y2

)

P0

(3.7.2)

An A = B = C = 0 tìte den mporoÔme na apofasÐsoume kai prèpei nadiereun soume n to shmeÐo P0 eÐnai shmeÐo akrottou ìqi eÐte phgaÐnontac seìrouc trÐthc txhc, eÐte me thn bo jeia grafik c parstashc. An A = B = 0kai C 6= 0 tìte an to C > 0 èqoume shmeÐo topikoÔ elaqÐstou, en¸ an C < 0èqoume shmeÐo topikoÔ megÐstou. An A = 0 en¸ B 6= 0 den mporoÔme naapofasÐsoume gia thn fÔsh tou shmeÐou P0.

'Estw t¸ra ìti A 6= 0. Tìte (apodeÐxte to)

f(x0 + h, y0 + k)− f(x0, y0) = 12A

[(Ah + Bk)2 − k2∆],∆ : = (B2 − AC) (3.7.3)

An B = C = 0 tìte to prìshmo thc diaforc f(x0 + h, y0 + k) − f(x0, y0)kajorÐzetai apo autì tou A se pl rh analogÐa me thn perÐptwsh tou C 6= 0parapnw kai ja èqoume topikì elqisto an A > 0 en¸ ja èqoume topikìmègisto an A < 0.Genik ja èqoume ta ex c apotelèsmata pou faÐnontai ston pÐnaka 3.1.



PerÐptwsh Sumpèrasma

∆ < 0 kai A > 0(C > 0) Topikì Elqisto

∆ < 0 kai A < 0(C < 0) Topikì Mègisto

∆ > 0 Sagmatikì shmeÐo

∆ = 0 Aprosdiìristo

PÐnakac 3.1: Qarakthrismìc akrottwn.

Otan lème sagmatikì shmeÐo, ennooÔme ìti en kinhjoÔme kat kpoiakateÔjunsh ja broÔme ìti èqoume shmeÐo topikoÔ megÐstou, en¸ kat llh jabroÔme ìti èqoume shmeÐo topikoÔ elaqÐstou. Pragmatik gia h, k tètoia ¸steAh+Bk = 0, dhl. kinoÔmenoi kat m koc thc eujeÐac A(x−x0)+B(y−y0) = 0ja isqÔei f(x0 + h, y0 + k) − f(x0, y0) < 0 (lìgw thc Ex. (3.7.3) ) opìte è-qoume topikì mègisto. Omwc gia h, k tètoia ¸ste k = 0, dhl. kat m koc thceujeÐac y = y0 èqoume f(x0 + h, y0 + k)− f(x0, y0) > 0 dhl. topikì elqisto.Qarakthristikì pardeigma sagmatikoÔ shmeÐou eÐnai to O(0, 0) sthn grafik parstash thc z = x2 − y2 (knte thn ston upologist sac).

Pardeigma 2.16: 'Estw z = 2x− x2 − y2 − 4y − 4. Ta krÐsima shmeÐaeÐnai aut gia ta opoÐa fx = 2 − 2x = 0 kai fy = −2y − 4 = 0 dhl. mìno toshmeÐo P (1,−2). EpÐshc A = −2, B = 0, C = −2. 'Ara ∆ = −2 < 0 kaiA < 0 kai èqoume shmeÐo topikoÔ megÐstou.

Pardeigma 2.17: An z = x2 − y2 tìte gia ta krÐsima shmeÐa eÐnaifx = 2x = 0, fy = −2y = 0. 'Ara èqoume to P0(0, 0). EpÐshc A = 2, B =0, C = −2 opìte ∆ = 4 > 0. EÐnai loipìn sagmatikì shmeÐo.

Pardeigma 2.18: An z = x2 + y2 tìte gia ta krÐsima shmeÐa eÐnai fx =2x = 0, fy = 2y = 0. 'Ara èqoume to P0(0, 0). EpÐshc A = 2, B = 0, C = 2opìte ∆ = −4 < 0 kai A = 2 > 0. EÐnai loipìn shmeÐo topikoÔ elaqÐstou.

Erqìmaste t¸ra sthn perÐptwsh pou zhtme akrìtatec timèc mÐac sunrthshcz = f(x, y) upì thn epiplèon sunj kh φ(x, y) = 0, dhl. se poiì shmeÐo thckampÔlhc tou epipèdou pou perigrfetai apì aut n thn exÐswsh èqoume akrì-tato. SqhmatÐzoume thn sunrthsh F (x, y) = f(x, y) + λφ(x, y) ìpou λ eÐnaio legìmenoc pollaplasiast c Lagrange. ProsdiorÐzoume t¸ra ta akrìtataapì tic

(∂F

∂x

)

P0

=

(∂F

∂y

)

P0

=

(∂F

∂λ

)

P0

= 0 (3.7.4)


3.8. ASKHSEIS 41

SqhmatÐzoume epÐshc thn orÐzousa

∆1 =

∣∣∣∣∣∣

Fxx, Fxy, φxFxy, Fyy, φyφx, φy, 0

∣∣∣∣∣∣P0

(3.7.5)

En ∆1 < 0 èqoume topikì elqisto en¸ en ∆1 > 0 èqoume topikì mègistosto shmeÐo P0.

Pardeigma 2.19: BreÐte ta akrìtata thc z = x + y pnw ston kÔklox2+y2 = 1. Ed¸ sqhmatÐzoume thn sunrthsh F (x, y) = x+y+λ(x2+y2−1).Ta akrìtata eÐnai anmesa sta shmeÐa ìpou Fx = 1 + 2λx = 0, Fy =1 + 2λy = 0 kai x2 + y2 − 1 = 0. EpilÔontac autì to sÔsthma brÐsk-oume tic lÔseic P0(x0, y0, λ0) = (

√2/2,

√2/2,−√2/2) kai P1(x1, y1, λ1) =

(−√2/2,−√2/2,√2). EpÐshc upologÐzontac touc ìrouc thc orÐzousac brÐsk-oume ìti (∆1)P0 = 4

√2 > 0 opìte èqoume topikì mègisto en¸ (∆1)P1 =

−4√2 < 0 ra èqoume topikì elqisto. MporeÐte na anazht sete kai ap'eujeÐac ta shmeÐa pnw ston monadiaÐo kÔklo pou to jroisma twn suntetag-mènwn touc eÐnai mègisto elqisto.

Pardeigma 2.20: BreÐte thn elqisth apìstash anmesa sthn eujeÐay + x = 2 kai to shmeÐo O(0, 0). Zhtme thn elqisth tim thc z = d(P, O) =√

x2 + y2 me thn sunj kh y+x−2 = 0. 'Ara F (x, y) =√

x2 + y2+λ(x+y−2)kai apì tic Fx = 2x + λ = 0, Fy = 2y + λ = 0, Fλ = x + y − 2 = 0 prokÔpteiìti to zhtoÔmeno shmeÐo eÐnai P0(x0, y0, λ0) = (1, 1,−2). EpÐshc (∆1)P0 =−√2 < 0 pou shmaÐnei ìti ìntwc prìkeitai gia elqisto me dmin =

√2.

3.8 Ask seic

1. BreÐte me bsh ton orismì tic parag¸gouc pr¸thc txhc twn paraktwsunart sewn sto upodeiknuìmeno shmeÐo.

z = ln(x2 + y2), P0(1, 0),

z = cos(xy), P0(0,π

2),

z =x− yx + y

, P0(0, 1),

z = xey, P0(1,−1).

2. BreÐte tic parag¸gouc pr¸thc kai deÔterhc txhc twn paraktw sunart -sewn:

z =x2

y2− x

y, z =

x− yx + y

, z =x

x2 + y2,



z = (ex+2y + y2)1/2, z = xy, z = Arctan(y

x),

w = (x2 + y2 + z2)1/2, w = ln(x2 + y2 + z2)

z = xsin(x + y) + ycos(x + y).

3. DeÐxte ìti h F (x, y) = (x − y)φ(x + y) epalhjeÔei thn exÐswsh (x −y)(Fx − Fy) − 2F = 0, ìpou φ eÐnai mÐa aujaÐreth sunrthsh. OmoÐwcdeÐxte ìti h F (x, y) = y2+xyln(xy) ikanopoieÐ thn yFxy−xFxx−Fx = 0.

4. DeÐxte ìti h sunrthsh thc Ex. (3.1.5) dèn eÐnai suneq c sthn arq O(0, 0), lìgw tou ìti to ìrio den uprqei, all par' ìla aut oi merikècpargwgoi wc prìc x, y uprqoun (me bsh ton orismì).

5. DeÐxte ìti h sunrthsh

f(x, y) = g(x) + g(y), g(t) =

{tsin(1/t), t 6= 0

0 t = 0

eÐnai suneq c sthn arq O(0, 0) (me bsh ton orismì thc sunèqeiac)all den uprqoun oi merikèc pargwgoi wc prìc x, y.


f(x, y) =

{x3

x2+y2, P 6= (0, 0)

0 P = (0, 0)

eÐnai suneq c sthn arq all to ìrio thc Ex. (3.1.6) den uprqei, dhl.den eÐnai diaforÐsimh.


f(x, y) =

{(x2 + y2)sin( 1√

x2+y2), P 6= (0, 0)

0 P = (0, 0)

eÐnai diaforÐsimh sthn arq , all oi merikèc thc pargwgoi wc prìc x, yden eÐnai suneqeÐc sunart seic ekeÐ, dhl. den eÐnai C1.

8. DeÐxte ìti h sunrthsh f(x, y) = Arctan(y/x) eÐnai armonik sunrthsh.To Ðdio kai gia thn sunrthsh dunamikoÔ f(x, y, z) = (1/r) = (x2 +y2 +z2)−1/2.

9. BreÐte thn sunj kh ¸ste h sunrthsh f(x, y) = sin(mx)sinh(ny) naeÐnai armonik stic dÔo diastseic. Ed¸ m,n eÐnai akèraioi kai sinhy =(ey − e−y)/2.


3.8. ASKHSEIS 43

10. ApodeÐxte to Je¸rhma tou Euler thc Ex. (3.2.4), paragwgÐzontac thnEx. (3.2.3) wc prìc λ. DeÐxte ìti eÐnai omogeneÐc oi paraktw sunart -seic

z = x3 + 3x2y + 2y3, z = (x2 + y2)ex2+y2

2xy

DeÐxte ìti mÐa omogen c sunrthsh bajmoÔ m ìpwc aut thc Ex. (3.2.3)ikanopoieÐ thn x2fxx + 2xyfxy + y2fyy = m(m− 1)f .

11. UpologÐste tic merikèc parag¸gouc wc prìc r, s twn paraktw sunart -sewn:

z = Arctan(x/y), x = rsins, y = rcoss

z = x2 + xy, x = rcoss, y = rsins

z = xy, x = rs, y = r/s

EpÐshc thn olik pargwgo wc prìc t gia tic sunart seic:

z = ln(x2 + y2), x = sint, y = cost

z = x/y, x = et, y = lnt

z = ex−2y, x = sint, y = t3

12. BreÐte thn y′(x) kai thn y′′(x) gia tic sunart seic pou orÐzontai upì thnpeplegmènh morf F (x, y) = 0, ìpou

x2 + y2 − 3xy = a, xtany + ytanx = bexy − x− 2y = 0, xy2 + 4xy3 + 2 = 0,

sin(xy)− exy − xy = 0ApodeÐxte thn Ex. (3.3.3).

13. ApodeÐxte thn Ex. (3.4.2). An (x, y) → (u, v) → (r, s) eÐnai mÐa seirmetasqhmatism¸n deÐxte ìti

D(x, y)

D(r, s)=

D(x, y)

D(u, v)

D(u, v)

D(r, s)

14. BreÐte tic iakwbianèc orÐzousec gia touc metasqhmatismoÔc apì tickartesianèc stic kulindrikèc kai sfairikèc suntetagmènec antÐstoiqa.EpalhjeÔste tic sqèseic metasqhmatismoÔ twn stoiqeiwd¸n ìgkwn.

15. DeÐxte ìti oi metablhtèc

u = u(x, y) = Arcsinx + Arcsiny, v = v(x, y) = x√

1− y2 + y√

1− x2

den eÐnai anexrthtec metaxÔ touc kai breÐte thn sqèsh pou tic sundèei.



16. Exetste an eÐnai anexrthtec oi paraktw metablhtèc kai an ìqi breÐtethn sqèsh pou tic sundèei.

u = 3x + 4y − zv = 2x− y + 3zw = 6x + 8y − 2z

f1 = x + y + z

f2 = x2 + y2 + z2

f3 = xy + yz + zx

17. BreÐte to olikì diaforikì gia tic sunart seic z = tan(y/x), z = xy,z = 1/

√x2 + y2.

18. UpologÐste proseggistik kai me thn bo jeia tou olikoÔ diaforikoÔ tictimèc

(1.04)2.02,√

27 · 3√

1021

f(1.98, 0.015), f(x, y) = ex2sin(xy)

19. Exetste an oi praktw sqèseic eÐnai tèleia diaforik kpoiwn sunart -sewn f(x, y) kai an nai breÐte tic.

(ey + yex)dx + (xey + ex)dy

(excosy − eysinx)dx + (eycosx− exsiny)dy(2xy + y2)dx + (2xy + x2)dy

(y +1

x)dx + (x +

1

y)dy

20. BreÐte to polu¸numo Taylor trÐthc txhc gia tic paraktw sunart seic,sta antÐstoiqa shmeÐa.

f(x, y) =1

1 + x + y, P0(0, 0)

f(x, y) = ln(1 + x2 + y2), P0(0, 0)

f(x, y) =x− y1 + y2

, P0(1, 2)

f(x, y) =1√

1 + x2 + y2, P0(1, 1)


3.8. ASKHSEIS 45

21. ParagwgÐste me bsh ton kanìna Leibniz to olokl rwma

f(x) =

∫ x2

0

Arctan(t

x)dt

UpologÐste to olokl rwma me bsh thn∫

Arctan(x/a) = xArctan(x/a)−(a/2)ln(x2 + y2) kai met paragwgÐste gia na katal xete sthn Ðdiasqèsh.

22. ParagwgÐzontac thn sqèsh∫ b0

dx1+ax

= 1aln(1+ab) wc prìc a upologÐste

to olokl rwma∫ b0

xdx(1+ax)2

.

23. UpologÐste me parag¸gish wc prìc thn parmetro a to olokl rwma

I(a) =

∫ ∞0

Arctan(ax)

x(1 + x2)

lambnontac up' ìyin ìti I(0) = 0.

24. BreÐte to eÐdoc twn krÐsimwn shmeÐwn kai ta akrìtata gia tic sunart -seic

z = 2x− x2 − y2 − 4y − 4,z = 3xy − x2y − xy2,z = xyln(x + y),

z = x3 + y3 − 9xy + 1,z = sinx + siny + sin(x + y)

z = x2 + y2 − y(x + a), (a > 0)z = x4 + y4 − 4xy + 1

25. BreÐte to mègisto thc parstashc V = xyz upì thn sunj khx + y + z = c, (c > 0).

26. BreÐte to mègisto thc parstashc F = xy2z3 upì thn sunj khx2 + y2 + z2 = δ2, (δ > 0).

27. BreÐte ta akrìtata twn snuart sewn upì tic antÐstoiqec sunj kec

z = x + y, x2 + y2 = 1

z = 6− 4x− 3y, x2

9+

y2

4= 1

z = xy, x + y = 1

28. BreÐte thn elqisth apìstash tou shmeÐou P0(0, 0) apì thn eujeÐax + y = a, (a > 0).



29. BreÐte thn elqisth apìstash tou shmeÐou P0(3, 1, 2) apì to epÐpedoz = 2x + 4y.

30. BreÐte thn elqisth apìstash thc eujeÐac y = x− 2 apì thn parabol y = x2.


Keflaio 4

Oloklhr¸mata

47

48 KEFALAIO 4. OLOKLHRWMATA

4.1 Dipl Oloklhr¸mata

'Estw D ènac kleistìc kai fragmènoc tìpoc sto epÐpedo. Ja melet soumeto diplì olokl rwma mÐac sunrthshc pou orÐzetai se èna tètoio tìpo gia naapofÔgoume thn eisagwg genikeumènwn oloklhrwmtwn pou eÐnai ektìc touskopoÔ twn shmei¸sewn aut¸n. 'Estw akìma f(x, y) sunrthsh orismènh kaifragmènh ston tìpo D. MporoÔme na qwrÐsoume ton tìpo se mikr orjog¸niatm mata me eujeÐec parllhlec proc touc xonec Ox, Oy. An {Dk/k =1, ..., n} eÐnai ta orjog¸nia aut kai |Dk| to embadìn enìc tètoiou kommatioÔlambnoume to ìrio

limn→∞

n∑

k=1

f(xk, yk)|Dk| = lim|D|→0n∑

k=1

f(xk, yk)|Dk|. (4.1.1)

Ed¸ Pk(xk, yk) eÐnai tuqaÐo shmeÐo pou an kei sto orjog¸nio Dk kai|D| = max{|Dk|/k = 1, ..., n}. An uprqei to ìrio autì, anexart twc toutrìpou me ton opoÐo epilègoume ta mikr aut orjog¸nia, tìte lème ìti hsunrthsh eÐnai oloklhr¸simh ston tìpo D kai to ìrio autì to sumbolÐzoumeme

I =

∫ ∫

D

f(x, y)dxdy (4.1.2)

H plèon endiafèrousa kathgorÐa sunart sewn pou eÐnai oloklhr¸simec eÐnaioi suneqeÐc sunart seic, me tic opoÐec ja asqolhjoÔme ed¸. UpenjumÐzoumeìti mÐa suneq c sunrthsh se èna kleistì kai fragmèno tìpo, ìpwc o D, eÐnaikai fragmènh s' autìn. To teleutaÐo fusik shmaÐnei ìti uprqei stajerM ∈ R tètoia ¸ste |f(x, y)| ≤ M .

H gewmetrik shmasÐa tou diploÔ oloklhr¸matoc eÐnai mesh. An qwrÐcperiorismì thc genikìthtac upojèsoume ìti ston tìpo D, h f(x, y) ≥ 0 tìteto diplì olokl rwma isoÔtai arijmhtik me ton ìgko (se kubikèc mondec pouorÐzontai apì tic mondec twn tri¸n orjokanonik¸n axìnwn) ktw apì thngrafik parstash thc fdhl. Vf =

∫ ∫D

f(x, y)dxdy. EpÐshc an f(x, y) = 1 tìte lambnoume ari-jmhtik to embadìn tou tìpou, ED =

∫ ∫D

dxdy.Oi stoiqei¸deic idiìthtec tou oloklhr¸matoc sunrthshc mÐac metablht c,

diathroÔntai kai ed¸ basik ametblhtec. DÔo piì shmantikèc idiìthtec eÐnaioi akoloujec:

∣∣∣∣∫ ∫

D

f(x, y)dxdy

∣∣∣∣ ≤∫ ∫

D

|f(x, y)|dxdy

m|D| ≤∫ ∫

D

f(x, y)dxdy ≤ M |D| (4.1.3)

ìpou m,M h elqisth kai mègisth tim thc sunrthshc antÐstoiqa kai |D| toembadìn tou tìpou.


4.1. DIPLA OLOKLHRWMATA 49

Gia na gryoume èna algìrijmo upologismoÔ tou diploÔ oloklhr¸matocja jewr soume ef' ex c kanonikoÔc tìpouc. Enac tìpoc ja eÐnai kanonikìc ankje eujeÐa parllhlh prìc touc xonec Ox, Oy tèmnei to sÔnoro tou tìpouse dÔo to polÔ shmeÐa. Kje tìpoc mporeÐ na analujeÐ se ènwsh kanonik¸ntìpwn. 'Enac kanonikìc tìpoc mporeÐ na grafeÐ kat touc akìloujouc dÔotrìpouc:

a ≤ x ≤ bφ1(x) ≤ y ≤ φ2(x) (4.1.4)

c ≤ y ≤ dg1(y) ≤ x ≤ g2(y) (4.1.5)

Tìte to diplì olok rwma upologÐzetai me èna apì touc akìloujouc dÔo isodÔ-namouc trìpouc

I =

∫ bx=a

dx

∫ φ2(x)y=φ1(x)

dyf(x, y) =

∫ dy=c

dy

∫ g2(y)x=g1(y)

dxf(x, y) (4.1.6)

Pardeigma 4.1: Na upologisteÐ to olokl rwma I =∫ ∫

D(x + y)dxdy

ston tìpo pou perikleÐetai apì tic kampÔlec y = x2, y = 13(4−x2) kai x = 2.

O tìpoc D mporeÐ na grafeÐ ¸c 1 ≤ x ≤ 2 kai 13(4− x2) ≤ y ≤ x2. SÔmfwna

me ta parapnw ja èqoume

I =

∫ 2x=1

dx

[∫ x2

y= 13(4−x2)

(x + y)dy

]=

∫ 2x=1

dx

[xy +

y2

2

]x2

y= 13(4−x2)

=

=

∫ 21

dx

[4

3x3 +

4

9x4 +

4

9x2 − 4

3x− 8

9

]' 5.88

EÐdame parapnw dÔo basikèc efarmogèc tou diploÔ oloklhr¸matoc. Thnqr sh tou ston upologismì ìgkou kai epifneiac. Epiplèon aut¸n mporoÔmena upologÐsoume kai to embadìn thc epifneiac z = f(x, y) pou problletaipnw ston tìpo D. DÐnetai apì thn sqèsh

Ef =

∫ ∫

D

√1 +

(∂f

∂x

)2+

(∂f

∂y

)2dxdy (4.1.8)

Pardeigma 4.2: UpologÐste thn epifneia thc sfaÐrac aktÐnac R. Jew-roÔme to pnw hmisfaÐrio pou paristnetai apì thn sunrthsh z =

√R2 − x2 − y2.

Tìte (∂f/∂x) = −x/√

R2 − x2 − y2 kai paromoÐwc gia thn merik pargwgo



wc prìc y. Tìte to zhtoÔmeno olokl rwma isoÔtai me

Ef =

∫ ∫

D

R√R2 − x2 − y2dxdy =

∫ 2π0

∫ R0

R√R2 − r2 rdrdθ =

= 2π

∫ R2

0

Rdu

2√

R2 − u = 2πR2

ìpou o tìpoc D pou oloklhr¸same eÐnai o kÔkloc aktÐnac R sto epÐpedoOxy. 'Ara to embadìn thc epifneiac thc sfaÐrac isoÔtai me 4πR2. Ed¸sthn ousÐa sunant same èna genikeumèno olokl rwma giatÐ h oloklhrwtèasunrthsh den eÐnai suneq c ston kleistì tìpo, all par' ìla aut, eÐdameìti to olokl rwma sugklÐnei.

4.2 Tripl Oloklhr¸mata

Me anlogo trìpo orÐzetai kai to triplì olokl rwma. En w = f(x, y, z)eÐnai sunrthsh orismènh kai fragmènh ston kleistì kai fragmèno tìpo V ⊂R3 tìte orÐzoume

I =

∫ ∫ ∫

V

f(x, y, z)dxdydz = limn→∞

n∑

k=1

f(xk, yk, zk)|∆Vk| =

= lim|∆V |→0

n∑

k=1

f(xk, yk, zk)|∆Vk| (4.2.1)

Ed¸ qwrÐsame ton tìpo V se tm mata {∆Vk/k = 1, ..., n} me epÐpeda parllh-la prìc ta suntetagmèna epÐpeda Oxy, Oyz, Ozx kai to shmeÐo Pk(xk, yk, zk)eÐnai opoiod pote shmeÐo tou kommatioÔ ∆Vk. Epiplèon knoume aut thndiamèrish tou tìpou V leptìterh paÐrnontac to ìrio n → ∞ isodÔnama|∆V | → 0 ìpou|∆V | = max{|∆Vk|/k = 1, ..., n}.

En mÐa sunrthsh f(x, y, z) eÐnai suneq c se kleistì kai fragmèno tìpoeÐnai kai oloklhr¸simh, kai apoteleÐ thn plèon qr simh klsh sunart sewn.Opwc kai sthn perÐptwsh tou diploÔ oloklhr¸matoc ènac kanonikìc tìpoc VmporeÐ na grafeÐ ¸c

a ≤ x ≤ bg1(x) ≤ y ≤ g2(x)

φ1(x, y) ≤ z ≤ φ2(x, y) (4.2.2)kai tìte o algìrijmoc gia ton upologismì tou triploÔ oloklhr¸matoc se ènatètoio tìpo eÐnai

I =

∫ bx=a

dx

∫ g2(x)y=g1(x)

dy

∫ φ2(x,y)z=φ1(x,y)

dz f(x, y, z) (4.2.3)


4.3. ALLAGH METABLHTWN SE POLLAPLA OLOKLHRWMATA 51

Se kje diadoqik olokl rwsh oi upìloipec metablhtèc jewroÔntai stajerèckaixekinme apì thn z−olokl rwsh.

En epilèxoume f(x, y, z) = 1 tìte to triplì olokl rwma sumpÐptei ari-jmhtik me ton ìgko tou tìpou V , ìpwc eÔkola mporoÔme na antilhfjoÔme.

Pardeigma 4.3: UpologÐste ton ìgko tou tetraèdrou pou orÐzetai apìta suntetagmèna epÐpeda Oxy, Oyz kai Ozx kai to epÐpedo me exÐswsh x +y + z = 1. Ed¸ o tìpoc V mporeÐ na ekfrasteÐ wc akoloÔjwc

0 ≤ x ≤ 10 ≤ y ≤ 1− x0 ≤ z ≤ 1− x− y

ìpwc mporoÔme eÔkola na diapist¸soume. 'Ara o zhtoÔmenoc ìgkoc eÐnai

I =

∫ 1x=0

dx

∫ 1−xy=0

dy

∫ 1−x−yz=0

dz =

∫ 1x=0

dx

∫ 1−xy=0

dy(1− x− y) =

=

∫ 1x=0

dx

[(1− x)y − y

2

2

]y=1−x

y=0

=

∫ 1x=0

dx

[(1− x)2 − (1− x)

2

2

]=

=1

2

∫ 1x=0

dx(1− x)2 = 16

4.3 Allag Metablht¸n se Pollapl Oloklhr¸-mata

Pollèc forèc o upologismìc enìc pollaploÔ oloklhr¸matoc eÐnai dÔskolocse kpoio sÔsthma suntetagmènwn (x, y, z). Endèqetai ìmwc an to metatrèy-oume se èna olokl rwma wc prìc kpoiec nèec metablhtèc, na upologÐzetaipolÔ eukolìtera. H metatrop aut gÐnetai mèsw twn Iakwbian¸n orizous¸n.'Estw

I =

∫ ∫

D

f(x, y)dxdy (4.3.1)

kai x = x(u, v), y = y(u, v) o metasqhmatismìc pou mac pei apì ton tìpo Dston opoÐo orÐzetai to olokl rwma, se èna llo tìpo D′ tou epipèdou (u, v).MporoÔme tìte na upologÐsoume to olokl rwma wc

I =

∫ ∫

D′f [x(u, v), y(u, v)]

D(x, y)

D(u, v)dudv =

∫ ∫

D′F (u, v)dudv (4.3.2)

ìpou h nèa sunrthsh prokÔptei apì ton metasqhmatismì ìlwn twn sunart -sewn tou pr¸tou oloklhr¸matoc sto epÐpedo (u, v).



OmoÐwc en

I =

∫ ∫ ∫

V

f(x, y, z)dxdydz (4.3.3)

eÐnai èna triplì olokl rwma kai x = x(u, v, w), y = y(u, v, w) kai z =z(u, v, w) ènac metasqhmatismìc pou mac pei apì ton tìpo V se èna tìpo V ′ ,tìte

I =

∫ ∫ ∫

V′f [x(u, v, w), y(u, v, w), z(u, v, w)]

D(x, y, z)

D(u, v, w)dudvdw =

=

∫ ∫ ∫

V ′F (u, v, w)dudvdw (4.3.4)

Pardeigma 4.4: Na upologÐsete to olokl rwma I =∫ ∫

D(x2+y2)dxdy

ìpou o tìpoc D eÐnai o kÔkloc aktÐnac R. Ed¸ aplopoioÔntai ta prgmataan pme stic sntetagmènec (r, θ) ìpou x = rcosθ, y = rsinθ. Tìte o neìctìpoc D′ ja eÐnai oD′: 0 ≤ r ≤ R, 0 ≤ θ ≤ 2π. Sunep¸c ja èqoume

I =

∫ ∫

D′r2

D(x, y)

D(r, θ)drdθ =

∫ 2πθ=0

∫ Rr=0

r3drdθ =πR4

2

4.4 Efarmogèc: Kèntra Mzac kai Ropèc Adrneiac

Ektìc apì tic kajar gewmetrikèc efarmogèc twn pollapl¸n oloklhrwmtwnpou eÐdame parapnw, uprqoun kai shmantikèc efarmogèc pou proèrqontaiapì thn Fusik . 'Estw D ènac epÐpedoc tìpoc, pou prosomoizei èna ulikìepÐpedo antikeÐmeno me epifaneiak puknìthta mzac σ(x, y) (se opoiesd potemondec, èstw gr/cm2). Tìte h sunolik tou mza kai thn jèsh tou kèntroumzac tou (x̄, ȳ) dÐnontai apì tic

M =

∫ ∫

D

σ(x, y)dxdy

x̄ =1

M

∫ ∫

D

xσ(x, y)dxdy

ȳ =1

M

∫ ∫

D

yσ(x, y)dxdy (4.4.1)

EpÐshc oi ropèc adrneiac tou wc prìc ton xona Oz kai touc xonec Ox kaiOy dÐnontai apì tic sqèseic

I0 =

∫ ∫

D

(x2 + y2)σ(x, y)dxdy

Ix =

∫ ∫

D

y2σ(x, y)dxdy


4.4. EFARMOGES: KENTRA MAZAS KAI ROPES ADRANEIAS 53

Iy =

∫ ∫

D

x2σ(x, y)dxdy (4.4.2)

Pardeigma 4.5: UpologÐste th jèsh tou kèntrou mzac kai tic ropècadrneiac gia èna orjog¸nio mzac M me pleurèc a, b. Upojètoume omogen katanom mzac, opìte ja èqoume σ(x, y) = M/ab. Tìte gia to kèntro mzacèqoume

x̄ =1

M

∫ ∫

D

xM

abdxdy =

1

ab

∫ a0

xdx

∫ b0

dy =a

2

ȳ =1

M

∫ ∫

D

xM

abdxdy =

1

ab

∫ a0

dx

∫ b0

ydy =b

2

Gia thn rop adrneiac ja èqoume

Ix =

∫ ∫

D

M

aby2dxdy =

M

ab

∫ a0

dx

∫ b0

y2dy =1

3Mb2

OmoÐwc brÐskoume ìti Iy = 13Ma2 kai I0 = 13M(a

2 + b2).Gia thn perÐptwsh swmtwn ston q¸ro, pou h puknìthta mzac touc dÐnetai

apì thn ρ(x, y, z) (se opoiesd pote mondec, èstw gr/cm3) ja èqoume ticantÐstoiqec genikeÔseic

M =

∫ ∫ ∫

V

ρ(x, y, z)dxdydz

x̄ =1

M

∫ ∫ ∫

V

xρ(x, y, z)dxdydz

ȳ =1

M

∫ ∫ ∫

V

yρ(x, y, z)dxdydz

z̄ =1

M

∫ ∫ ∫

V

zρ(x, y, z)dxdydz (4.4.3)

H rop adrneiac wc prìc ton xona Oz ja dÐnetai apì thn

Iz =

∫ ∫ ∫

V

(x2 + y2)ρ(x, y, z)dxdydz (4.4.4)

kai apì antÐstoiqec sqèseic gia touc llouc dÔo xonec, me kuklik enallag twn anexrthtwn metablht¸n (x, y, z).

Pardeigma 4.6: UpologÐste thn rop adrneiac mÐac sfaÐrac aktÐnac Rkai mzac M wc prìc ton xona Oz. Upojètontac omogen sfaÐra ja èqoumeρ(x, y, z) = ρ̄ = 3M/4πR3. Tìte

Iz = ρ̄

∫ ∫(x2 + y2)dxdy

∫ z=+√R2−(x2+y2)z=−

√R2−(x2+y2)

dz =



= 2ρ̄

∫ ∫(x2 + y2)

√R2 − (x2 + y2)dxdy =

= 4πρ̄

∫ R0

r2√

R2 − r2rdr =

= 2πρ̄

∫ R2

0

u√

R2 − udu = 25MR2

4.5 Ask seic

1. UpologÐste ta paraktw dipl oloklhr¸mata stouc antÐstoiqouc tìpoucD.

∫ ∫

D

(x2 + 2y)dxdy, D = {(x, y)/0 ≤ x ≤ 1, 0 ≤ y ≤ 1}∫ ∫

D

x2

y2dxdy, D = {(x, y)/1 ≤ x ≤ 2, 1

x≤ y ≤ x}

∫ ∫

D

xydxdy, D = {(x, y)/y2 ≤ x, x2 ≤ y}∫ ∫

D

exdxdy, D = {(x, y)/0 ≤ x ≤ 1, 0 ≤ y ≤ x}∫ ∫

D

√R2 − x2 − y2dxdy, D = {(x, y)/x2 + y2 ≤ R2}

2. Na upologÐsete to olokl rwma∫ ∫

D

(a− x− y)dxdy,

ston tìpo D pou periorÐzetai apì tic kampÔlec 3x + y = a, 32x + y = a

kai y = 0.

3. OmoÐwc to

I =

∫ ∫

D

dxdy,

ston tìpo pou periorÐzetai apì tic kampÔlec x+y = a kai x2 +y2 = a2.

4. Na upologÐsete to

I =

∫ 43

dx

∫ 21

dy

(x + y)2


4.5. ASKHSEIS 55

5. Na upologÐsete to olokl rwma

I =

∫ ∫ ∫

V

xyzdxdydz

ìpou o tìpoc V periorÐzetai apì ta suntetagmèna epÐpeda Oxy, Oyz,Ozx kai to epÐpedo x + y + z = 1.

6. Na upologÐsete to olokl rwma

I =

∫ ∫ ∫

V

√x2 + y2 + z2dxdydz

entìc thc sfaÐrac aktÐnac R.

7. Na upologÐsete to

I =

∫ 10

∫ 10

∫ 10

dxdydz√x + y + z + 1

8. UpologÐste ton ìgko tou stereoÔ pou frssetai apì to suntetagmènoepÐpedo Oxy, ton kÔlindro x2+y2 = a2 kai to paraboloeidèc z = x2+y2.

9. UpologÐste ton ìgko tou stereoÔ anmesa sta epÐpeda y = 0, z = 0,y = x kai ton kÔlindro x2 + z2 = a2.

10. OmoÐwc ton ìgko pou periorÐzetai apì ta epÐpeda x+y+z = a, 3x+y =a, 3

2x + y = a kai ta sunetagmèna epÐpeda y = 0, z = 0.

11. UpologÐste me katllhlh allag metablht¸n to

I =

∫ ∫

D

(x2 + y2)dxdy

ìpou o tìpoc D periorÐzetai apì tic kampÔlec x2− y2 = 1, x2− y2 = 9,xy = 2 kai xy = 4.

12. UpologÐste to zhtoÔmeno thc 'Askhshc 9 me katllhlh allag metabl-ht¸n sto epÐpedo Oxz.

13. UpologÐste me katllhlh allag metablht¸n touc ìgkouc thc sfaÐrackai tou elleiyoeidoÔc.

14. OmoÐwc to

I =

∫ ∫

D

√x2 + y2dxdy

mèsa ston kÔklo x2 + y2 = R2.



15. UpologÐste to embadìn thc èlleiyhc.

16. BreÐte to embadìn pou apokìptoun ta suntetagmèna epÐpeda Oxy, Oyz,Ozx apì to epÐpedo x

a+ y

b+ z

c= 1.

17. BreÐte to embadìn thc epifneiac z = 1− x2 − y2 pnw apì to epÐpedoOxy.

18. BreÐte to embadìn tou kulÐndrou x2 + y2 = 2ax metaxÔ tou epipèdouOxy kai tou k¸nou z2 = x2 + y2.

19. UpologÐste thn rop adrneiac gia rbdo m kouc L kai mzac M pouperistrèfetai gÔrw apì xona kjeto sto èna kro thc.

20. UpologÐste thn jèsh tou kèntrou mzac tou tìpou D pou periorÐzetaiapì tic kampÔlec y2 ≤ 4x, y2 ≤ 5 − x, y ≥ 0. Upojèste omogen katanom mzac.

21. UpologÐste tic ropèc adrneiac enìc orjogwnÐou mzac M kai pleur¸na, b wc prìc touc treÐc xonec.

22. UpologÐste ta kèntra mzac twn tìpwn pou periorÐzontai apì tic akìlou-jec kampÔlec: D1 : x + y = a, x2 + y2 = a2 kai D2 : y = x2, y =

√x.

23. UpologÐste th jèsh tou kèntrou mzac kai tic ropèc adrneiac gia kuk-likì tomèa aktÐnac R kai gwnÐac 2Φ me epifaneiak puknìthta σ = kr,ìpou k stajer kai r h aktinik apìstash sto epÐpedo.


Keflaio 5

Dianusmatik Anlush

57

58 KEFALAIO 5. DIANUSMATIKH ANALUSH

5.1 Dianusmatikèc sunart seic mÐac metabl-ht c

MÐa sqèsh thc morf c

~a(t) = a1(t)x̂ + a2(t)ŷ + a3(t)ẑ = (a1(t), a2(t), a3(t)) (5.1.1)

orÐzei gia kje tim thc pragmatik c metablht c t ∈ [a, b] èna dinusma stonq¸ro. OrÐzetai ètsi mÐa dianusmatik sunrthsh mÐac metablht c. Kaj¸c tot metablletai, genik to kro tou dianÔsmatoc diagrfei ston q¸ro mÐa kam-pÔlh C. MporoÔme na efarmìsoume ìlec tic gnwstèc prxeic pou aforoÔn sesunart seic mÐac metablht c, epÐshc kai se mÐa dianusmatik sunrthsh. 'EtsimporoÔme na prosjèsoume kai na afairèsoume dÔo tètoiec sunart seic, kaj¸ckai na paragwgÐsoume kai na oloklhr¸soume dianusmatikèc sunart seic. Wcpardeigma

~v(t) =d~a(t)

dt=

da1(t)

dtx̂ +

da2(t)

dtŷ +

da3(t)

dtẑ (5.1.2)

To dinusma ~v(t) èqei thn idìthta na eÐnai efaptìmeno se kje shmeÐo thckampÔlhc C kai onomzetai dinusma thc taqÔthtac. Epeid to stoiqei¸decm koc thc kampÔlhc, ìpwc autì metriètai me bsh kpoio aujaÐreto shmeÐoekkÐnhshc, eÐnai

ds =

√(da1dt

)2+

(da2dt

)2+

(da3dt

)2dt (5.1.3)

prokÔptei ìti

~v(t) =d~a(t)

dt=

d~a

ds

ds(t)

dt= v̂(s)

ds(t)

dt(5.1.4)

ìpou to v̂(s) = d~a/ds eÐnai to monadiaÐo efaptomenikì dinusma (dianusmatikìpedÐo) se tuqaÐo shmeÐo thc kampÔlhc kai ds(t)

dteÐnai to mètro thc taqÔthtac.

Pardeigma 5.1: 'Estw kinhtì pnw se kÔklo aktÐnac R pou perifèretaime gwniak taqÔthta ω. Tìte h dianusmatik sunrthsh pou parakoloujeÐto kinhtì se kje shmeÐo thc troqic tou eÐnai

~a(t) = Rcos(ωt)x̂ + Rsin(ωt)ŷ

ìpou ed¸ h parmetrìc mac t eÐnai o qrìnoc, kai θ = ωt h gwnÐa pou sqhmatÐzeih aktÐna tou kÔklou pou parakoloujeÐ to kinhtì me ton xona x. Tìte htaqÔthta eÐnai

~v(t) =d~a(t)

dt= −Rωsin(ωt)x̂ + Rωcos(ωt)ŷ


5.2. DIANUSMATIKES SUNARTHSEIS POLLWN METABLHTWN 59

To mètro thc taqÔthtac eÐnai |~v(t)| = (ds(t)/dt) = Rω ìpwc autì prokÔpteiapì thn parapnw sqèsh. Prgmati autì epibebai¸netai kai apì to gegonìcìti to m koc tou tìxou pnw se kÔklo aktÐnac R eÐnai s = Rθ = Rωt opìte(ds/dt) = Rω. To monadiaÐo efaptìmeno dinusma, ìpwc prokÔptei apì thnÐdia sqèsh eÐnai

v̂(t) =~v(t)

|~v(t)| = −sin(ωt)x̂ + cos(ωt)ŷ

kai me antikatstash prokÔptei mesa ìti ikanopoieÐtai h Ex. (5.1.4).To dinusma

~γ(t) =d~v(t)

dt=

d2~a(t)

dt2(5.1.5)

onomzetai dinusma epitqunshc. Sqetikì me ta dianusmatik pedÐa mÐackampÔlhc C, eÐnai to trÐakmo Frenet (Parrthma II).

Pardeigma 5.2: 'Estw ~a(t) = 2costx̂ + 2sintŷ + 3tẑ. BreÐte ta dianÔs-mata taqÔthtac kai epitqunshc kaj¸c kai ta mètra touc. Ja èqoume

~v(t) = −2sintx̂ + 2costŷ + 3ẑ~γ(t) = −2costx̂ + 2sintŷ|~v(t)| =

√13

|~γ(t)| = 2

5.2 Dianusmatikèc sunart seic poll¸n metabl-ht¸n

'Opwc kai prohgoumènwc mÐa sqèsh thc morf c

~F (x, y) = P (x, y)x̂ + Q(x, y)ŷ (5.2.1)

ìpou oi anexrthtec metablhtèc (x, y) orÐzontai se kpoio tìpo D ⊂ R2 orÐzeimÐa dianusmatik sunrthsh dÔo metablht¸n sto epÐpedo. EpÐshc h sqès

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MAJHMATIKAIIteachers.cm.ihu.gr/anastasiou/wp-content/uploads/2012/02/notes... · xonec. Oi arijmoÐ (»;·;‡) apoteloÔn tic sunist¸sec tou monadiaÐou dianÔs-matoc ˆr. To eswterikì